Chapter 4
Pure Bending
INTRODUCTION
Bending Stress
W
W
A
A
B
L
Bending of Members made of Several Materials
30
Aluminum
5
0.5 TYP
Brass
Steel
2.5
24
Brass
Aluminum
2.5
12
1.5
Cross-section, in
Cross-section, in
Stress Concentrations
r
r
M
M
D
d
M
M
D
d
2r
Introduction
4-1
Eccentric Axial Loading in a Plane of Symmetry
P
Unsymmetric Bending
My
My
My
Mx
Mx
Mx
4-2
Introduction
BENDING STRESS
L
Bending Stress
4-3
BENDING STRESS- continued
y
z
σx = −
4-4
y
x
z
x
My
I
Bending Stress
SECTION MODULUS
y
−σ 2
c2
z
c1
x
σ1
y
h
x
b
σx = −
Section Modulus
My
M
=−
I
S
4-5
Example
Find the maximum bending stress at section a-a (3 m from A) of the
W150x29.8 beam. Units: N, m.
4000
4000
4000
a
B
A
A
a
1.5
6
1.5
W150x29.8
Area, A = 3790mm
Depth, d = 157 mm
Flange Width, b f = 153mm
Flange Thickness, t f = 9.3mm
Web Thickness, t w = 6.6mm
2
I x = 17.2 x106 mm
4
I y = 5.56 x106 mm
S x = 219 x103mm
4
3
S y = 72.7 x103mm
4-6
3
Example
Find the bending stresses at the wall at points A and B for a 6" pipe
with a wall thickness of 0.125". Units: lb, ft.
2000
A
B
3
Cross-section
4-7
Example
Find the bending stresses at the wall at points A and B for the W6x20
beam. Units: lb, ft.
2000
A
B
3
Cross-section
W6x20
Area, A = 5.87in
Depth, d = 6.20in
Flange Width, b f = 6.02in
Flange Thickness, t f = 0.365in
Web Thickness, t w = 0.260in
2
I x = 41.4in
I y = 13.3in
4
4
S x = 13.4in
3
S y = 4.41in
4-8
3
Example
Find the bending stresses at the wall at points A and B for the C6x13
beam. Units: lb, ft.
200
A
3
B
Cross-section
C6x13
Area, A = 3.83in
Depth, d = 6.00in
Flange Width, b f = 2.16in
Flange Thickness, t f = 0.343in
Web Thickness, t w = 0.437in
2
I x = 17.4in
I y = 1.05in
4
4
S x = 5.80in
3
S y = 0.642in
3
x = 0.514in
4-9
Example
Find the maximum bending stress at section a-a (3 m from A) of the
W150x29.8 beam. Units: N/m, m.
200
a
200
A
A
B
a
6
W150x29.8
Area, A = 3790mm
Depth, d = 157 mm
Flange Width, b f = 153mm
Flange Thickness, t f = 9.3mm
Web Thickness, t w = 6.6mm
2
I x = 17.2 x106 mm
4
I y = 5.56 x106 mm
S x = 219 x103mm
4
3
S y = 72.7 x103mm
4-10
3
Example
The sign is subjected to a wind force of 250 lb at it’s centroid 7' from
the center of the column. The column has an outside diameter of 10"
and a wall thickness of 0.25". Considering only this force, determine
the maximum bending stress. Units: ft.
7
ENGINEERING
WAY
NEXT EXIT
30
A
Front View
Side View
4-11
PARALLEL-AXIS THEOREM
MOMENTS OF INERTIA
PARALLEL-AXIS THEOREM
y
I x = ∑ ( I x′ + Ad y2 )
I y = ∑ ( I y′ + Ad x2 )
x
EXAMPLE
t (uniform thk.)
a/2
x
x
a/2
b
4-12
Parallel-Axis Theorem
Example
The simply-supported beam below has a cross-sectional area as
shown. Determine the bending stress that acts at points c and d,
located at section a-a (3 m from A). Units: N/m, mm (uno).
5000
5000
A
a
A
B
a
6m
20
c
150
15
20
150
250
d
4-13
Example
The member is designed to resist a moment of 5 kip•in about the
horizontal axis. Determine the maximum normal stress in the member
for the two similar cross-sections. Units: in.
1.5
3
0.5
0.25
4-14
Example
Compare the bending stresses between the two cases for a moment
about the horizontal axis. Case 1 is a simple solid cross-section,
whereas case 2 is made up of 3 identical boards. The 3 boards aren’t
connected together and simply rest on one another. Units: in.
1.5
3
Case 1
1.5
3
Case 2
4-15
Example
The two beams are connected by a thin rigid plate on the top and
bottom side of the flanges. Find the bending stresses at the wall at
points A, B and C for the W6x20 beam. Units: lb, ft
2000
A, C
A
C
B
B
3
Cross-section
W6x20
Area, A = 5.87in
Depth, d = 6.20in
Flange Width, b f = 6.02in
Flange Thickness, t f = 0.365in
Web Thickness, t w = 0.260in
2
I x = 41.4in
I y = 13.3in
4
4
S x = 13.4in
3
S y = 4.41in
4-16
3
Example
The two beams are connected by a thin rigid plate on the top and
bottom side of the flanges. Find the bending stresses at the wall at
points A, B and C for the W6x20 beam. Units: lb, ft
A, C
A
C
B
2000
2000
B
3
Cross-section
W6x20
Area, A = 5.87in
Depth, d = 6.20in
Flange Width, b f = 6.02in
Flange Thickness, t f = 0.365in
Web Thickness, t w = 0.260in
2
I x = 41.4in
I y = 13.3in
4
4
S x = 13.4in
3
S y = 4.41in
3
4-17
Example
The two beams are connected by bolts through the flanges. Find the
bending stresses at the wall at points A and B for the W6x20 beam.
Units: lb, ft
2000
A
B
A
3
B
Cross-section
W6x20
Area, A = 5.87in
Depth, d = 6.20in
Flange Width, b f = 6.02in
Flange Thickness, t f = 0.365in
Web Thickness, t w = 0.260in
2
I x = 41.4in
I y = 13.3in
4
4
S x = 13.4in
3
S y = 4.41in
4-18
3
Example
The two beams are connected by bolts through the flanges. Find the
bending stresses at the wall at points A and B for the W6x20 beam.
Units: lb, ft
A
A
2000
2000
B
3
B
Cross-section
W6x20
Area, A = 5.87in
Depth, d = 6.20in
Flange Width, b f = 6.02in
Flange Thickness, t f = 0.365in
Web Thickness, t w = 0.260in
2
I x = 41.4in
I y = 13.3in
4
4
S x = 13.4in
3
S y = 4.41in
3
4-19
Example
The simply-supported beam below has a cross-sectional area as
shown. Determine the bending stress that acts at points c and d,
located at section a-a (3 m from A). Units: N/m, mm (UNO).
5000
5000
A
a
A
a
6m
B
150
30
c
100
d
24
Cross-section
4-20
BENDING OF MEMBERS MADE OF SEVERAL MATERIALS
y
y
N.A.
εx
σx
σ x = −n
Bending of Members Made of Several Materials
My
I
4-21
Example
Find the stress in each of the three metals if a moment of 12 k-in is
applied about the horizontal axis.
E (aluminum)= 10E6, E(steel)= 30E6, E (brass)= 15E6 psi. Units: in.
Aluminum
0.5 TYP
Brass
2.5
Steel
Brass
Aluminum
1.5
4-22
Example
A W150x29.8 wide flange beam is reinforced with wood planks that are
securely connected to the flanges. Esteel/Ewood= 20. If the allowable
stresses in the wood and steel are 4.5 MPa and 52 MPa, respectively,
determine the allowable distributed load w based on section a-a (3 m
from A). Units: N/m, mm (UNO).
153
w
w
a
40
A
Cross-section
A
B
a
6m
W150x29.8
Area, A = 3790mm
Depth, d = 157 mm
Flange Width, b f = 153mm
Flange Thickness, t f = 9.3mm
Web Thickness, t w = 6.6mm
2
I x = 17.2 x106 mm
4
I y = 5.56 x106 mm
S x = 219 x103mm
4
3
S y = 72.7 x103mm
3
4-23
Example
Two steel plates are securely fastened to a 6"x10" wood beam.
Esteel/Ewood= 20. Knowing that the beam is bent about the horizontal
axis by a 125 kip-in moment, determine the maximum stress in (a) the
wood, (b) the steel. Units: in.
2 TYP
0.375 TYP
Cross-section
4-24
Example
Determine the stress in the concrete and steel if a moment of 1500
kip-in is applied about the horizontal axis. Area of steel= 3.14 sq. in.
E (steel)= 30E6 psi, E (concrete)= 3.75E6 psi. Units: in.
30
5
24
2.5
12
Cross-section
30
Cross-section
4-25
Example
Determine the required steel area for the beam to be balanced.
Allowable stress in the steel and concrete are 33,000 and 3,000 psi
respectively.
E (steel)= 29E6 psi, E (concrete)= 3.5E6 psi. Units: in.
22
8
Cross-section
Cross-section
4-26
STRESS CONCENTRATIONS
r
r
M
M
M
d
D
d
D
σ max
⎛ My ⎞
=k⎜
⎟
⎝ I ⎠
3.0
2r
3.0
r
D/d= 3
2.8
2.6
M
d
D
2.0
1.8
2
2.6
2.2
k
1.1
1.6
M
M
d
D
1.5
1.2
2.4
1.5
1.2
2.2
r
2.8
M
2
2.4
k
M
2r
1.1
2.0
1.8
1.6
1.4
1.0
0
0.05
0.10
0.15
1.05
1.4
1.02
1.01
1.2
1.2
0.20
0.25
0.30
r/d
Stress-concentration factors for flat
bars with fillets under pure bending
1.0
0
0.05
0.10
0.15
0.20
0.25
0.30
r/d
Stress-concentration factors for flat
bars with groves under pure bending
Ref.: W.D. Pilkey, Peterson’s Stress Concentration Factors, 2nd ed., John Wiley and Sons, New York, 1997
Stress Concentrations
4-27
Example
For the 13 mm thick plate, determine the largest bending moment that
can be applied if the allowable bending stress is 90 MPa. Units: mm.
R6
M
M
100
150
3.0
2.6
M
M
2
2.4
d
D
1.5
1.2
2.2
k
r
D/d= 3
2.8
2.0
1.8
1.1
1.6
1.4
1.02
1.01
1.2
1.0
0
0.05
0.10
0.15
0.20
0.25
0.30
r/d
Stress-concentration factors for flat
bars with fillets under pure bending
Ref.: W.D. Pilkey, Peterson’s Stress Concentration
Factors, 2nd ed., John Wiley and Sons, New York, 1997
4-28
Example
For the 7/8" thick plate, determine the largest bending moment that can
be applied if the allowable bending stress is 24,000 psi. Units: in.
R0.375
M
M
4
6
.75
3.0
r
2.8
2
2.6
M
2.2
d
D
1.5
1.2
2.4
k
M
2r
1.1
2.0
1.8
1.6
1.05
1.4
1.2
1.0
0
0.05
0.10
0.15
0.20
0.25
0.30
r/d
Stress-concentration factors for flat
bars with groves under pure bending
Ref.: W.D. Pilkey, Peterson’s Stress Concentration
Factors, 2nd ed., John Wiley and Sons, New York, 1997
4-29
ECCENTRIC AXIAL LOADING IN A PLANE OF SYMMETRY
P
P
M
In general,
σx =
4-30
P My
+
A I
Eccentric Axial Loading in a Plane of Symmetry
Example
For the solid rectangular bar, determine the largest load P that can be
applied based on a maximum normal stress of 130 MPa. Ignore any
stress concentrations. Units: mm.
P
a
P
P
b
a
b
100
50
125
Section a-a
125
Section b-b
P
4-31
Example
The three loads are applied at the end of the W6x20 beam. Find the
normal stress at the wall at point A for the beam, (a) if all three loads
are applied, (b) the bottom load is removed. Units: lb, ft.
A
20000
A
20000
20000
3
Cross-section
W6x20
Area, A = 5.87in
Depth, d = 6.20in
Flange Width, b f = 6.02in
Flange Thickness, t f = 0.365in
Web Thickness, t w = 0.260in
2
I x = 41.4in
I y = 13.3in
4
4
S x = 13.4in
3
S y = 4.41in
4-32
3
Example
The 100 mm diameter solid circular bar has an eccentric load P
applied. Determine the maximum location x that the load can be
placed without inducing any tensile stresses. Units: mm.
x
P
4-33
Example
Compute the maximum tension and compression stresses located at
section a-a. Units: N, mm (UNO).
2m
400
a
125
B
A
a
6m
8 (uniform)
100
60
Section a-a
400
a
125
A
a
4m
4-34
UNSYMMETRIC BENDING
My
σx = −
Mz
σx =
My
Mz y Myz
+
Iz
Iy
P Mz y M yz
−
+
A
Iz
Iy
My
My
Mz
Mz
Mz
Unsymmetric Bending
4-35
Example
For the W6x20 section, determine the normal stresses at A and B.
Units: kip•in.
60
B
A
45
W6x20
Area, A = 5.87in
Depth, d = 6.20in
Flange Width, b f = 6.02in
Flange Thickness, t f = 0.365in
Web Thickness, t w = 0.260in
2
I x = 41.4in
I y = 13.3in
4
4
S x = 13.4in
3
S y = 4.41in
4-36
3
Example
For the WT18x150 section, determine the normal stresses at A, B and
C. Units: k•in.
700
A
B
1500
C
WT18x150
Area, A = 44.10in
Depth, d = 18.4in
Flange Width, b f = 16.7in
Flange Thickness, t f = 1.68in
Web Thickness, t w = 0.945in
2
I x = 1230in
I y = 648in
4
4
S x = 86.1in
3
S y = 77.8in
3
y = 4.13in
4-37
Example
For the channel section, determine the normal stresses at A and B.
Units: kN•m, mm.
A
10
15°
220
20 TYP
B
12
100
4-38
Example
For the L76x76x12.7 angle section, determine the normal stresses at A
and B. Units: N•m.
A
400
Area, A = 1770mm
d = b = 76mm
x = y = 23.6mm
Thickness, t = 12.7 mm
2
I x = I y = 0.915 x106 mm
4
rz = 14.8mm
B
4-39
SUMMARY
Bending Stress
W
σx = −
A
B
My
M
=−
I
S
L
Bending of Members made of Several Materials
30
Aluminum
5
0.5 TYP
Brass
Steel
2.5
σ x = −n
Brass
My
I
24
Aluminum
2.5
12
1.5
Cross-section, in
Cross-section, in
Stress Concentrations
r
r
M
M
M
D
M
d
D
d
3.0
M
M
2
2.4
d
D
2r
1.5
1.2
2.2
k
r
D/d= 3
2.8
2.6
2.0
1.8
1.1
1.6
1.4
1.02
1.01
1.2
1.0
0
0.05
0.10
0.15
0.20
0.25
0.30
r/d
Stress-concentration factors for flat
bars with fillets under pure bending
4-40
Summary
SUMMARY
Eccentric Axial Loading in a Plane of Symmetry
P
σx =
P My
−
A I
Unsymmetric Bending
My
My
My
Mz
Mz
Mz
σx =
Summary
P Mz y M yz
−
+
A
Iz
Iy
4-41