Fourth Edition
CHAPTER
5
MECHANICS OF
MATERIALS
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T. DeWolf
Lecture Notes:
J. Walt Oler
Texas Tech University
Analysis and Design
of Beams for Bending
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Analysis and Design of Beams for Bending
Introduction
Shear and Bending Moment Diagrams
Sample Problem 5.1
Sample Problem 5.2
Relations among Load, Shear, and Bending Moment
Sample Problem 5.3
Sample Problem 5.5
Design of Prismatic Beams for Bending
Sample Problem 5.8
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Introduction
• Objective  Analysis and design of beams.
• Beams  structural members supporting loads
at various points along the member.
• Transverse loadings of beams are classified as
concentrated loads or distributed loads.
• Applied loads result in internal forces consisting
of a shear force (from the shear stress
distribution) and a bending couple (from the
normal stress distribution).
• Normal stress is often the critical design criteria
x  
My
I
m 
Mc M

I
S
Requires determination of the location and
magnitude of largest bending moment.
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Introduction
Classification of Beam Supports
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Shear and Bending Moment Diagrams
• Determination of maximum normal and
shearing stresses requires identification of
maximum internal shear force and bending
couple.
• Shear force and bending couple at a point are
determined by passing a section through the
beam and applying an equilibrium analysis on
the beam portions on either side of the
section.
• Sign conventions for shear forces V and V’
and bending couples M and M’
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Beer • Johnston • DeWolf
Sample Problem 5.1
SOLUTION:
• Treating the entire beam as a rigid
body, determine the reaction forces.
For the timber beam and loading
shown, draw the shear and bendmoment diagrams and determine the
maximum normal stress due to
bending.
• Section the beam at points near
supports and load application points.
Apply equilibrium analyses on
resulting free bodies to determine
internal shear forces and bending
couples.
• Identify the maximum shear and
bending moment from plots of their
distributions.
• Apply the elastic flexure formulas to
determine the corresponding
maximum normal stress.
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Sample Problem 5.1
SOLUTION:
• Treating the entire beam as a rigid body, determine
the reaction forces
from
 Fy  0   M B : RB  46 kN RD  14 kN
• Section the beam and apply equilibrium analyses
on resulting free bodies
 Fy  0
 20 kN  V1  0
V1  20 kN
 M1  0
20 kN0 m   M1  0
M1  0
 Fy  0
 20 kN  V2  0
V2  20 kN
 M2  0
20 kN2.5 m   M 2  0
M 2  50 kN  m
V3  26 kN
M 3  50 kN  m
V4  26 kN M 4  28 kN  m
V5  14 kN M 5  28 kN  m
V6  14 kN M 6  0
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Sample Problem 5.1
• Identify the maximum shear and bending
moment from plots of their distributions.
Vm  26 kN M m  M B  50 kN  m
• Apply the elastic flexure formulas to
determine the corresponding
maximum normal stress.
S  16 b h 2  16 0.080 m 0.250 m 2
 833 .33 10  6 m3
MB
50 103 N  m
m 

S
833 .33 10  6 m3
 m  60.0 10 6 Pa
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Beer • Johnston • DeWolf
Sample Problem 5.2
SOLUTION:
• Replace the 45-kN load with an
equivalent force-couple system at D.
Find the reactions at B by considering
the beam as a rigid body.
• Section the beam at points near the
support and load application points.
Apply equilibrium analyses on
resulting free bodies to determine
internal shear forces and bending
couples.
The structure shown is constructed of a
W250  167 rolled-steel beam. (a)
Draw the shear and bending-moment
diagrams for the beam and the given
loading. (b) Determine normal stress in
• Apply the elastic flexure formulas to
sections just to the right and left of point
determine the maximum normal
D.
stress to the left and right of point D.
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Beer • Johnston • DeWolf
Sample Problem 5.2
SOLUTION:
• Replace the 45-kN load with equivalent forcecouple system at D. Find reactions at B.
• Section the beam and apply equilibrium
analyses on resulting free bodies.
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Sample Problem 5.2
• Apply the elastic flexure formulas to
determine the maximum normal stress to
the left and right of point D.
From Appendix C for a W250  167 rolled
steel shape, S = 208  106mm about the
XX axis.
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Relations among Load, Shear, and Bending Moment
• Relationship between load and shear:
 Fy  0 : V  V  V   w x  0
V   w x
dV
 w
dx
xD
VD  VC    w dx
xC
• Relationship between shear and bending
moment:
 M C  0 :
M  M   M  V x  wx x  0
M  V x  12 w x 
2
2
dM
V
dx
M D  MC 
xD
 V dx
xC
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Sample Problem 5.3
SOLUTION:
• Taking the entire beam as a free body,
determine the reactions at A and D.
• Apply the relationship between shear and
load to develop the shear diagram.
Draw the shear and bendingmoment diagrams for the beam
and loading shown.
• Apply the relationship between bending
moment and shear to develop the bendingmoment diagram.
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Sample Problem 5.3
SOLUTION:
• Taking the entire beam as a free body, determine the
reactions at A and D.
• Apply the relationship between shear and load to
develop the shear diagram.
dV
 w
dx
dV   w dx
 zero slope between concentrated loads
 linear variation over uniform load segment
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Sample Problem 5.3
• Apply the relationship between bending
moment and shear to develop the bendingmoment diagram.
dM
V
dx
dM  V dx
 bending moment at A and E is zero
 bending moment variation between A, B,
C, and D is linear
 bending moment variation between D
and E is quadratic
 net change in bending moment is equal to
areas under shear distribution segments
 total of all bending-moment changes across
the beam should be zero
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Beer • Johnston • DeWolf
Sample Problem 5.5
SOLUTION:
• Taking the entire beam as a free body,
determine the reactions at C.
• Apply the relationship between shear
and load to develop the shear diagram.
Draw the shear and bending-moment
diagrams for the beam and loading
shown.
• Apply the relationship between
bending moment and shear to develop
the bending moment diagram.
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Beer • Johnston • DeWolf
Sample Problem 5.5
SOLUTION:
• Taking the entire beam as a free body,
determine the reactions at C.
 Fy  0   12 w0 a  RC
a

 M C  0  12 w0 a L    M C
3

RC  12 w0 a
a

M C   12 w0 a L  
3

Results from integration of the load and shear
distributions should be equivalent.
• Apply the relationship between shear and load
to develop the shear diagram.
a
2 
 
x
 x
VB  V A    w0 1   dx    w0  x  

a
2a 
 
0 
0
a
VB   12 w0 a    area under load curve
 No change in shear between B and C.
 Compatible with free body analysis
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Beer • Johnston • DeWolf
Sample Problem 5.5
• Apply the relationship between bending moment
and shear to develop the bending-moment
diagram.
a
a
2 


 x 2 x3 
x


M B  M A    w0  x   dx   w0   

 2 6a 

2a  

0


 0
M B   13 w0a 2
L


M B  M C    12 w0 a dx   12 w0 aL  a 
a
a w0 
a
M C   16 w0 a3L  a  
L 
2 
3
Results at C are compatible with free-body
analysis
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MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Design of Prismatic Beams for Bending
• The largest normal stress is found at the surface where the
maximum bending moment occurs.
M max c M max
m 

I
S
• A safe design requires that the maximum normal stress be
less than the allowable stress for the material used. This
criteria leads to the determination of the minimum
acceptable section modulus.
 m   all
S min 
M max
 all
• Among beam section choices which have an acceptable
section modulus, the one with the smallest weight per unit
length or cross-sectional area will be the least expensive
and the best choice.
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Beer • Johnston • DeWolf
Sample Problem 5.8
SOLUTION:
• Considering the entire beam as a free
body, determine the reactions at A and
D.
A simply supported steel beam is to
carry the distributed and concentrated
loads shown. Knowing that the
allowable normal stress for the grade
of steel to be used is 160 MPa, select
the wide-flange shape that should be
used.
• Develop the shear diagram for the
beam and load distribution. From the
diagram, determine the maximum
bending moment.
• Determine the minimum acceptable
beam section modulus. Choose the
best standard section which meets this
criteria.
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Sample Problem 5.8
• Considering the entire beam as a free body,
determine the reactions at A and D.
 M A  0  D5 m   60 kN1.5 m   50 kN4 m 
D  58.0 kN
 Fy  0  Ay  58.0 kN  60 kN  50 kN
Ay  52.0 kN
• Develop the shear diagram and determine the
maximum bending moment.
V A  Ay  52.0 kN
VB  VA  area under load curve  60 kN
VB  8 kN
• Maximum bending moment occurs at
V = 0 or x = 2.6 m.
M max  area under shear curve, A to E 
 67.6 kN
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Beer • Johnston • DeWolf
Sample Problem 5.8
• Determine the minimum acceptable beam
section modulus.
M max 67.6 kN  m
S min 

 all
160 MPa
 422.5 10  6 m3  422.5 103 mm3
Shape
S  103 mm 3
W410  38.8
637
W360  32.9
474
W310  38.7
549
W250  44.8
535
W200  46.1
448
• Choose the best standard section which meets
this criteria.
W 360 32.9
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