Transcript 8 - Benvenuti nel sito web del Gruppo di Costruzione di

Fourth 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
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
Fourth
Edition
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
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Introduction
• 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
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 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
Mc
I
8-4
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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 xy
near the surface where x is also large.
• max may be greater than m
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
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
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
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
VA  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
a 
y
90.4 mm
σb   a b  117.2 MPa 
c
103 mm
 102.9 MPa
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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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 mm 3
 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 
 159.9 MPa  150 MPa 
Design specification is not satisfied.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
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 all = 24 ksi
and 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.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.2
SOLUTION:
• Determine reactions at A and D.
 M A  0  RD  59 kips
 M D  0  RA  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
Smin 

 119.7 in 3
 all
24 ksi
select W21  62 beam section
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Sample Problem 8.2
• Find maximum shearing stress.
Assuming uniform shearing stress in web,
 max 
Vmax
43 kips

 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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