8_Principal stresses.pdf

MECHANICS OF
MATERIALS
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. DeWolf
David F. Mazurek
Sanjeev Sanghi
Lecture Notes:
Brock E. Barry
U.S. Military Academy
Sanjeev Sanghi
Indian Institute of Technology Delhi
CHAPTER
Copyright © 2015 McGraw-Hill Education. Permission required for reproduction or display.
Seventh Edition in SI Units
8
Principle Stresses
Under a Given
Loading

Seventh
Edition
in
SI
Units
MECHANICS OF MATERIALS Beer • Johnston • DeWolf • Mazurek • Sanghi
Contents
8 - 2
Introduction
Principal Stresses in a Beam
Sample Problem 8.1
Sample Problem 8.2
Design of Transmission Shafts
Sample Problem 8.3
Stresses Under Combined Load
Sample Problem 8.5

Seventh
Edition
in
SI
Units
Introduction
8 - 3
• In Chapter 1 and 2, you learned how to determine the normal stress due to
centric loads.
• In Chapter 3, you analyzed the distribution of shearing stresses in a circular
member due to a twisting couple.
• In Chapter 4, you determined the normal stresses caused by bending couples.
• In Chapters 5 and 6, you evaluated the shearing stresses due to transverse
loads.
• In Chapter 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.

Seventh
Edition
in
SI
Units
8 - 4
• Prismatic beam subjected to transverse
loading
x m
xy m
My Mc
I I
VQ VQ
It It
 
 
  
  
• Can the maximum normal stress within
the cross-section be larger than
m
Mc
I
 
• Principal stresses determined from methods
of Chapter 7
Fig. 8.4 Transversely loaded
prismatic beam.
Fig. 8.5 Stress elements at selected
points of a beam.
Fig. 8.6 Principal stress elements at
selected points of a beam.

Seventh
Edition
in
SI
Units
8 - 5
Fig. 8.7 Narrow rectangular
cantilever beam supporting a single
concentrated load.
Fig. 8.9 Stress trajectories in a
rectangular cantilevered beam
supporting a single concentrated load.
Figure 8.8 Distribution of principal stresses in two transverse
sections of a rectangular cantilever beam supporting a single
concentrated load.

Seventh
Edition
in
SI
Units
8 - 6
• When the width of the cross section varies so that
large shearing stresses xy occur at points near the
surface, the principal stress max may be larger than
m.
Fig. 8.10 Key stress analysis
locations in I-shaped beams.

Seventh
Edition
in
SI
Units
Sample Problem 8.1
8 - 7
A 160-kN force is applied at the end
of a W200x52 rolled-steel beam.
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’.
SOLUTION:
• Determine shear and bending
moment in Section A-A’
• Calculate the normal stress at top
surface and at flange-web junction.
• Evaluate the shear stress at flange-
web junction.
• Calculate the principal stress at
flange-web junction

Seventh
Edition
in
SI
Units
Sample Problem 8.1
8 - 8
SOLUTION:
• Determine shear and bending moment in
Section A-A’
  
160kN 0.375m 60kN-m
160kN
A
A
M
V
 

• Calculate the normal stress at top surface
and at flange-web junction.
 
6 3
60kN m
511 10 m
117.4MPa
90.4mm
117.4MPa
103mm
103.0MPa
A
a
b
b a
M
S
y
c

 


 


 

Fig. 1 Free-body diagram of beam, with
section at A-A’
Fig. 2 Cross-section dimensions and
normal stress distribution.

Seventh
Edition
in
SI
Units
Sample Problem 8.1
8 - 9
• Calculate the principal stress at
flange-web junction
 
 
 
2 2
1 1
max 2 2
2
2
103.0 103.0
96.5
2 2
160.9MPa 150 MPa
b b b
   
  
 
  
 
 
 
Design specification is not satisfied.
• Evaluate shear stress at flange-web junction.
 
  
  
3 3
6 3
6 3
6 4
206 12.6 96.7 251.0 10 mm
251.0 10 m
160kN 251.0 10 m
52.9 10 m 0.00787m
96.5MPa
A
b
Q
V Q
It




   
 

 


Fig. 3 Dimensions to evaluate Q at point
b.
Fig. 4 Stress element for coordinate and
principal orientations at point b; Mohr’s
circle for point b.

Seventh
Edition
in
SI
Units
Sample Problem 8.2
8 - 10
The overhanging beam AB supports a
uniformly distributed load and a
concentrated load. Knowing that
for the grade of steel to be used
all = 165 MPa and all = 100 MPa,
select the wide-flange beam which
should be used.
SOLUTION:
• Determine reactions at A and D.
• Find maximum shearing stress.
• Find maximum normal stress.
• Calculate required section modulus
and select appropriate beam section.
• Determine maximum shear and
bending moment from shear and
bending moment diagrams.

Seventh
Edition
in
SI
Units
Sample Problem 8.2
8 - 11
• Calculate required section modulus
and select appropriate beam section.
6 3
max
min
323.2kN m
1.959 10 mm
165MPa
select W530 92 beam section (lightest available shape)
all
M
S


   

• Determine maximum shear and bending
moment from shear and bending moment
diagrams.
max
max
323.2kN m with 54.9 kN
193.5kN
M V
V
  

SOLUTION:
• Determine reactions at A and D.
0 265.5kN
0 184.5kN
A D
A
D
M R
M R
  
  


Fig. 1 Free-body diagram of beam;
shear and bending moment diagrams.

Seventh
Edition
in
SI
Units
Sample Problem 8.2
8 - 12
• Find maximum shearing stress.
Assuming uniform shearing stress in web,
max
max 2
193.5 kN
35.6 MPa 100MPa
5436.6 mm
web
V
A
    
• Find maximum normal stress.
 
max
6 3
b 6 2
323.2kN m
156.1MPa
2.07 10 m
(250.9 mm)
156.1MPa 147MPa
(266.5 mm)
54.9 kN
10.1MPa
5436.6 10 m
a
b
b a
web
M
S
y
c
V
A

 




  

  
  

 
2
2
max
147MPa 147MPa
10.1MPa
2 2
147.7MPa 165MPa OK

 
  
 
 
  
Fig. 3 Key stress analysis locations and
normal stress distribution.
Fig. 4 The stress element at point b and
the Mohr’s circle for point b.
Fig. 2 I-shape cross section properties.

Seventh
Edition
in
SI
Units
8 - 13
• If power is transferred to and from the
shaft by gears or sprocket wheels, the
shaft is subjected to transverse loading
as well as torsional 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.
Fig. 8.11 Loading on gear-shaft systems. (a) Forces
applied to gear teeth. (b) Free-body diagram of shaft,
with gear forces replaced by equivalent force-couple
systems applied to shaft.

Seventh
Edition
in
SI
Units
8 - 14
• Shaft section requirement,
 
2 2
max
min all
M T
J
c 

 

 
 
• Maximum shearing stress,
 
2 2 2
2
max
2 2
max
2 2
for a circular or annular cross-section, 2
m
m
Mc Tc
I J
I J
c
M T
J

 

     
   
   
     
 

 
• At any section,
2 2 2
where
m y z
m
Mc
M M M
I
Tc
J


  

Fig. 8.12 (a) Torque and bending
couples acting on shaft cross section. (b)
Bending couples replaced by their
resultant M.
Fig. 8.14 Mohr’s circle for shaft loading.

Seventh
Edition
in
SI
Units
Sample Problem 8.3
8 - 15
Solid shaft AB 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 all = 50 MPa, determine the
smallest permissible diameter for the
shaft.
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.
• Calculate minimum allowable shaft
diameter.

Seventh
Edition
in
SI
Units
Sample Problem 8.3
8 - 16
SOLUTION:
• Determine the gear torques and corresponding
tangential forces.
 
 
 
30kW
597N m
2 2 8Hz
597N m
3.73kN
0.16m
20kW
398N m 6.63kN
2 8Hz
10kW
199N m 2.49kN
2 8Hz
E
E
E
E
C C
D D
P
T
f
T
F
r
T F
T F
 


   

  
   
   
• Find reactions at A and B.
0.932kN 6.22kN
2.80kN 2.90kN
y z
y z
A A
B B
 
 
Fig. 1 Free-body diagram of shaft AB
and its gears.
Fig. 2 Free-body diagram of shaft AB, with
gear forces replaced by equivalent force-
couple systems.

Seventh
Edition
in
SI
Units
Sample Problem 8.3
8 - 17
• Identify critical shaft section from torque and
bending moment diagrams.
 
2 2 2 2 2 2
max
1160 373 597
1357N m
y z
M M T
    
 
Fig. 3 Analysis of free-body diagram of shaft AB alone with equivalent force-couple loads is
equivalent to superposition of bending moments from vertical loads, horizontal loads, and applied
torques.

Seventh
Edition
in
SI
Units
Sample Problem 8.3
8 - 18
• Calculate minimum allowable shaft diameter.
2 2 2
6 3
1357 N m
27.14 10 m
50MPa
y z
all
M M T
J
c 

 


  
2 51.7 mm
d c
 
3 6 3
27.14 10 m
2
0.02585m 25.85m
J
c
c
c
 
  
 
For a solid circular shaft,
Fig. 4 Bending moment
components and torque at critical
section.

Seventh
Edition
in
SI
Units
8 - 19
• 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.
Fig. 8.15 Member ABDE subjected to
several forces.
Fig. 8.16 Free-body diagram of segment
ABC to determine the internal forces and
moments at cross section C.

Seventh
Edition
in
SI
Units
8 - 20
• 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.
Fig. 8.17 Internal forces and couple vectors
separated into (a) those causing normal
stresses and (b) those causing shearing
stresses.
Fig. 8.18 Normal and shearing stress at
points H and K.

Seventh
Edition
in
SI
Units
8 - 21
• 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.
1. The stresses involved must not exceed
the proportional limit of the material.
2. The deformations due to one of the
loadings must not affect the determination
of stresses due to the others.
3. The section used in your analysis must
not be too close to the points of application
of the given forces.
Fig. 8.19 Elements at points H and K
showing combined stresses.
Fig. 8.20 Elements at points H and K
showing principal stresses.

Seventh
Edition
in
SI
Units
Sample Problem 8.5
8 - 22
Three forces are applied to a short
steel post as shown. Determine the
principal stresses, principal planes and
maximum shearing stress at point H.
SOLUTION:
• Determine internal forces in Section
EFG.
• Calculate principal stresses and
maximum shearing stress.
Determine principal planes.
• Evaluate shearing stress at H.
• Evaluate normal stress at H.

Seventh
Edition
in
SI
Units
Sample Problem 8.5
8 - 23
SOLUTION:
• Determine internal forces in Section EFG.
     
  
30 kN 50kN 75kN
50kN 0.130m 75kN 0.200m
8.5kN m
0 30kN 0.100m 3kN m
x z
x
y z
V P V
M
M M
    
 
  
   
Note: Section properties,
  
  
  
3 2
3 6 4
1
12
3 6 4
1
12
0.040m 0.140m 5.6 10 m
0.040m 0.140m 9.15 10 m
0.140m 0.040m 0.747 10 m
x
z
A
I
I



  
  
  
Fig. 1 Equivalent force-couple system at
section containing points E, F, G, and H.

Seventh
Edition
in
SI
Units
Sample Problem 8.5
8 - 24
• Evaluate normal stress at H.
  
  
 
3 2 6 4
6 4
3kN m 0.020m
50kN
5.6 10 m 0.747 10 m
8.5kN m 0.025m
9.15 10 m
8.93 80.3 23.2 MPa 66.0MPa
z x
y
z x
M a M b
P
A I I

 

   

 
 



   
• Evaluate shearing stress at H.
    
  
  
1 1
6 3
6 3
6 4
0.040m 0.045m 0.0475m
85.5 10 m
75kN 85.5 10 m
9.15 10 m 0.040m
17.52MPa
z
yz
x
Q A y
V Q
I t




 
   
 

 


Fig. 2 Dimensions and bending couples
used to determine normal stresses.
Fig. 3 Dimensions and forces used to
determine the transverse shearing stress.

Seventh
Edition
in
SI
Units
Sample Problem 8.5
8 - 25
• Calculate principal stresses and maximum
shearing stress.
Determine principal planes.
2 2
max
max
min
p p
33.0 17.52 37.4MPa
33.0 37.4 70.4MPa
33.0 37.4 7.4MPa
17.52
tan2 2 27.96
33.0
13.98
p
R
OC R
OC R
CY
CD



 

   
    
     
   
 
max
max
min
37.4MPa
70.4MPa
7.4MPa
13.98
p






 
 
Fig. 4 Mohr’s circle at point H used for
finding principal and maximum shearing
stress and their orientation.

8_Principal stresses.pdf

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