Fourth Edition
CHAPTER
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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Beer • Johnston • DeWolf
Introduction
• In Chapters 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.
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MECHANICS OF MATERIALS
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Principle Stresses in a Beam
• Prismatic beam subjected to transverse
loading
My
Mc
σm =
I
I
VQ
VQ
τ xy = −
τm =
It
It
σx = −
• Principal stresses determined from methods
discussed in Chapter 7
• Determine if the maximum normal
stress within the cross-section is larger
than
Mc
σm =
I
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
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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 τxy
near the surface where σx is also large.
• σmax may be greater than σm .
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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 W200 × 52 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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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
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
σa =
90.4 mm
y
σb = σ a b = (117.2 MPa )
103 mm
c
= 102.9 MPa
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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
τb =
=
It
52.7 × 10− 6 m 4 (0.0079 m )
(
)
= 95.5 MPa
• Calculate the principal stress at
flange-web junction.
σ max = 12 σ b +
(12 σ b )2 + τ 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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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 be used σall = 165
MPa and τall = 100 MPa, select the
wide-flange 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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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.
• Calculate required section modulus
and select appropriate beam section.
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Beer • Johnston • DeWolf
Sample Problem 8.2
• Find maximum shearing stress.
Assuming uniform shearing stress in web,
• Find maximum normal stress.
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