Lecture Mechanics of materials (Third edition) - Chapter 8: Principle stresses under a given loading

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<b>MECHANICS OF </b>

<b>MATERIALS</b>

<b>Ferdinand P. Beer</b>

<b>E. Russell Johnston, Jr.</b>
<b>John T. DeWolf</b>

<b>Lecture Notes:</b>
<b>J. Walt Oler</b>

<b>Texas Tech University</b>

CHAPTER

Principle Stresses

Under a Given

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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

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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

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Principle Stresses in a Beam

• Prismatic beam subjected to transverse
loading

<i>It</i>
<i>VQ</i>
<i>It</i>

<i>VQ</i> <i>I</i>

<i>Mc</i>
<i>I</i>

<i>My</i>

<i>m</i>

<i>xy</i>

<i>m</i>
<i>x</i>

=
−

=

=
−

=

τ
τ

σ
σ

• Principal stresses determined from methods
of Chapter 7

• Can the maximum normal stress within
the cross-section be larger than

<i>I</i>
<i>Mc</i>
<i>m</i> =

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Principle Stresses in a Beam

• Cross-section shape results in large values of

τ

<i>_xy</i>
near the surface where

σ

<i>_x</i> is also large.

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Sample Problem 8.1

SOLUTION:

• Determine shear and bending
moment in Section <i>A-A’</i>

• 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.
Neglecting the effects of fillets and
of stress concentrations, determine

• Evaluate the shear stress at
flange-web junction.

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Sample Problem 8.1

SOLUTION:

• Determine shear and bending moment in
Section <i>A-A’</i>

(

)(

)

kN
160
m

-kN
60
m
375
.
0
kN
160
=
=
=
<i>A</i>
<i>A</i>
<i>V</i>
<i>M</i>

• Calculate the normal stress at top surface
and at flange-web junction.

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Sample Problem 8.1

• Evaluate shear stress at flange-web junction.

(

)

(

)

(

)

(

)

(

)

MPa
5
.
95
m
0079
.
0
m
10
7
.
52
m
10
6
.
248
kN
160
m
10
6
.
248
mm
10
6

.
248
7
.
96
6
.
12
204
4
6
3
6
3
6
3
3
=
ì
ì
=
=
ì
=
ì
=
ì
=

<i>It</i>
<i>Q</i>
<i>V</i>
<i>Q</i>
<i>A</i>
<i>b</i>

ã Calculate the principal stress at
flange-web junction

( )

(

)

9
.
102
9
.
102 2
2
2
2
1
2
1
max
⎞
⎛
+
+

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Sample Problem 8.2

The overhanging beam supports a
uniformly distributed load and a

concentrated load. Knowing that for
the grade of steel to used

σ

<i>_all</i> = 24 ksi
and

τ

<i>_all</i> = 14.5 ksi, select the
wide-flange beam which should be used.

SOLUTION:

• Determine reactions at <i>A</i> and <i>D.</i>

• Find maximum shearing stress.
• Find maximum normal stress.

• Calculate required section modulus
and select appropriate beam section.
• Determine maximum shear and

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