MECHANICS OF
MATERIALS
Third Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
John T.DeWolf
Lecture Notes:
J. Walt Oler
Texas Tech University
CHAPTER
©2002 The McGrawHill Companies, Inc. All rights reserved.
Analysis and Design
of Beams for Bending
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 2
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
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 3
Introduction
•
Bea
ms

structural members supporting loads at
various points along the member
•
Objective 
Analysis and design of beams
•
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
S
M
I
c
M
I
My
m
x
=
=
−
=
σ
σ
Requires determinati
on of the location and
magnitude of largest bending moment
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 4
Introduction
Classification of Beam Supports
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 5
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’
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 6
Sample Problem 5.1
For the timber beam and loading
shown, draw the shear and bend
moment diagrams and determine the
maximum normal stress due to
bending.
SOLUTION:
•
Treating the entire beam as a rigid
body, determine the reaction forces
•
Identify the maximum shear and
bendingmoment from plots of their
distributions.
•
Section the beam at points near
supports and load application points.
Apply equilibrium analyses on
resulting freebodies to determine
internal shear forces and bending
couples
•
Apply the elastic flexure formulas to
determine the corresponding
maximum normal stress.
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 7
Sample Problem 5.1
SOLUTION:
•
Treating the entire beam as a rigid body, determine
the reaction forces
∑
∑
=
=
=
=
kN
14
kN
40
:
0
from
D
B
B
y
R
R
M
F
•
Section the beam and apply equilibrium analyses
on resulting freebodies
()
(
)
0
0
m
0
kN
20
0
kN
20
0
kN
20
0
1
1
1
1
1
=
=
+
∑
=
−
=
=
−
−
∑
=
M
M
M
V
V
F
y
()
(
)
m
kN
50
0
m
5
.
2
kN
20
0
kN
20
0
kN
20
0
2
2
2
2
2
⋅
−
=
=
+
∑
=
−
=
=
−
−
∑
=
M
M
M
V
V
F
y
0
kN
14
m
kN
28
kN
14
m
kN
28
kN
26
m
kN
50
kN
26
6
6
5
5
4
4
3
3
=
−
=
⋅
+
=
−
=
⋅
+
=
+
=
⋅
−
=
+
=
M
V
M
V
M
V
M
V
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 8
Sample Problem 5.1
•
Identify the maximum shear and bending
moment from plots of their distributions.
m
kN
50
kN
26
⋅
=
=
=
B
m
m
M
M
V
•
Apply the elastic flexure formulas to
determine the corresponding
maximum normal stress.
(
)
(
)
3
6
3
3
6
2
6
1
2
6
1
m
10
33
.
833
m
N
10
50
m
10
33
.
833
m
250
.
0
m
080
.
0
−
−
×
⋅
×
=
=
×
=
=
=
S
M
h
b
S
B
m
σ
Pa
10
0
.
60
6
×
=
m
σ
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 9
Sample Problem 5.2
The structure shown is constructed of a
W10x112 rolledsteel beam. (a) Draw
the shear and bendingmoment diagrams
for the beam and the given loading. (b)
determine normal stress in sections just
to the right and left of point D.
SOLUTION:
•
Replace the 10 kip load with an
equivalent forcecouple 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 freebodies to determine
internal shear forces and bending
couples.
•
Apply the elastic flexure formulas to
determine the maximum normal
stress to the left and right of point D.
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 10
Sample Problem 5.2
SOLUTION:
•
Replace the 10 kip load with equivalent force
couple system at D. Find reactions at B.
•
Section the beam and apply equilibrium
analyses on resulting freebodies.
()
()
ft
kip
5
.
1
0
3
0
kips
3
0
3
0
:
2
2
1
1
⋅
−
=
=
+
∑
=
−
=
=
−
−
∑
=
x
M
M
x
x
M
x
V
V
x
F
C
to
A
From
y
()
(
)
ft
kip
24
96
0
4
24
0
kips
24
0
24
0
:
2
⋅
−
=
=
+
−
∑
=
−
=
=
−
−
∑
=
x
M
M
x
M
V
V
F
D
to
C
From
y
()
ft
kip
34
226
kips
34
:
⋅
−
=
−
=
x
M
V
B
to
D
From
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 11
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 W10x112 rolled
steel shape, S
= 126 in3
about the XX
axis.
3
3
in
126
in
kip
1776
:
in
126
in
kip
2016
:
⋅
=
=
⋅
=
=
S
M
D
of
right
the
To
S
M
D
o
f
le
ft
the
To
m
m
σ
σ
ksi
0
.
16
=
m
σ
ksi
1
.
14
=
m
σ
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 12
Relations Among Load, Shear, and Bending Moment
(
)
x
w
V
x
w
V
V
V
Fy
∆
−
=
∆
=
∆
−
∆
+
−
=
∑
0
:
0
∫
−
=
−
−
=
D
C
x
x
C
D
dx
w
V
V
w
dx
dV
•
Relationship between load and shear:
()
()
2
2
1
0
2
:
0
x
w
x
V
M
x
x
w
x
V
M
M
M
M
C
∆
−
∆
=
∆
=
∆
∆
+
∆
−
−
∆
+
=
∑
′
∫
=
−
=
D
C
x
x
C
D
dx
V
M
M
dx
dM
0
•
Relationship between shear and bending
moment:
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 13
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 bending
moment diagrams for the beam
and loading shown.
•
Apply the relationship between bending
moment and shear to develop the bending
moment diagram.
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 14
Sample Problem 5.3
SOLUTION:
•
Taking the entire beam as a free body, determine the
reactions at A
and D.
(
)
(
)
(
)
()
(
)
()
(
)
kips
18
kips
12
kips
26
kips
12
kips
20
0
0
F
kips
26
ft
28
kips
12
ft
14
kips
12
ft
6
kips
20
ft
24
0
0
y
=
−
+
−
−
=
=
∑
=
−
−
−
=
=
∑
y
y
A
A
A
D
D
M
•
Apply the relationship between shear and load to
develop the shear diagram.
dx
w
dV
w
dx
dV
−
=
−
=

zero slope between concentrated loads

linear variation over uniform load segment
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 15
Sample Problem 5.3
•
Apply the relationship between bending
moment and shear to develop the bending
moment diagram.
dx
V
dM
V
dx
dM
=
=

bending moment at A
and E
is zero

total of all bending moment changes across
the beam should be zero

net change in bending moment is equal to
areas under shear distribution segments

bending moment variation between D
and E
is quadratic

bending moment variation between A, B,
C and D
is linear
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 16
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.
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 17
Sample Problem 5.5
SOLUTION:
•
Taking the entire beam as a free body,
determine the reactions at C.
⎟
⎠
⎞
⎜
⎝
⎛
−
−
=
+
⎟
⎠
⎞
⎜
⎝
⎛
−
=
=
∑
=
+
−
=
=
∑
3
3
0
0
0
2
1
0
2
1
0
2
1
0
2
1
a
L
a
w
M
M
a
L
a
w
M
a
w
R
R
a
w
F
C
C
C
C
C
y
Results from integration of the load and shear
distributions should be equivalent.
•
Apply the relationship between shear and load
to develop the shear diagram.
()
curve
load
under
area
a
w
V
a
x
x
w
dx
a
x
w
V
V B
a
a
A
B
−
=
−
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
∫
⎟
⎠
⎞
⎜
⎝
⎛
−
−
=
−
0
2
1
0
2
0
0
0
2
1

No change in shear between B
and C.

Compatible with free body analysis
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 18
Sample Problem 5.5
•
Apply the relationship between bending moment
and shear to develop the bending moment
diagram.
2
0
3
1
0
3
2
0
0
2
0
6
2
2
a
w
M
a
x
x
w
dx
a
x
x
w
M
M
B
a
a
A
B
−
=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
∫
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
=
−
(
)
()
()
⎟
⎠
⎞
⎜
⎝
⎛
−
=
−
−
=
−
−
=
∫
−
=
−
3
2
3
0
0
6
1
0
2
1
0
2
1
a
L
w
a
a
L
a
w
M
a
L
a
w
dx
a
w
M
M
C
L
a
C
B
Results at C
are compatible with freebody
analysis
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 19
Design of Prismatic Beams for Bending
•
The largest normal stress is found at the surface where the
maximum bending moment occurs.
S
M
I
c
M
m
max
max
=
=
σ
•
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.
al
l
all
m
M
S
σ
σ
σ
max
min
=
≤
•
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.
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 20
Sample Problem 5.8
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 wideflange shape that should be
used.
SOLUTION:
•
Considering the entire beam as a free
body, determine the reactions at A
and
D.
•
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.
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 21
Sample Problem 5.8
•
Considering the entire beam as a freebody,
determine the reactions at A
and D.
()
(
)
(
)
(
)
(
)
kN
0
.
52
kN
50
kN
60
kN
0
.
58
0
kN
0
.
58
m
4
kN
50
m
5
.
1
kN
60
m
5
0
=
−
−
+
=
=
∑
=
−
−
=
=
∑
y
y
y
A
A
A
F
D
D
M
•
Develop the shear diagram and determine the
maximum bending moment.
()
kN
8
kN
60
kN
0
.
52
−
=
−
=
−
=
−
=
=
B
A
B
y
A
V
curve
load
under
area
V
V
A
V
•
Maximum bending moment occurs at
V
= 0 or x
= 2.6 m.
(
)
kN
6
.
67
,
max
=
=
E
to
A
curve
shear
under
area
M
©2002 The McGrawHill Companies, Inc. All rights reserved.
ME
C
HANI
CS
O
F MATERIAL
S
Third
Edition
Beer •Johnston •DeWolf
5 22
Sample Problem 5.8
•
Determine the minimum acceptable beam
section modul
us.
3
3
3
6
max
min
mm
10
5
.
422
m
10
5
.
422
MPa
160
m
kN
6
.
67
×
=
×
=
⋅
=
=
−
all
M
S
σ
•
Choose the best standard section which meets
this criteria.
448
1
.
46
W200
535
8
.
44
W250
549
7
.
38
W310
474
9
.
32
W360
637
38.8
W410
mm
,
3
×
×
×
×
×
S
Shape
9
.
32
360
×
W
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