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CE 234
STRENGTH OF MATERIALS
CHAPTER 4
Pure Bending
Lecture By: Dr. Özgür KÖYLÜOĞLU, Yeditepe University
Book: Mechanics of Materials, 6th Edition, by Ferdinand P. Beer; E. Russel
Johnston, Jr.; John T. DeWolf, David F. Mazurek, Mc Graw Hill.
Presentation Reference: Lecture Notes by J. Walt Oler, Texas Tech University on
Mechanics of Materials, 3rd Edition, by Ferdinand P. Beer; E. Russel Johnston,
Jr.; John T. DeWolf, Mc Graw Hill.
Pure Bending
400 N
400 N
300 mm
1500 mm 300 mm
40 N
120 N-m
40 N
120 N-m
2
Other Loading Types
3
Symmetric Member in Pure Bending
Fx    x dA  0
M y   z x dA  0
M z    y x dA  M
4
Bending Deformations
5
Strain Due to Bending
L=ρθ
At y from Neutral Axis;
L¢ = ( r - y)q
d = L '- L = ( r - y)q - rq = -yq
ex =
d
=
-yq
=-
y
(strain varies linearly)
L rq
r
Maximum absolute value of the strain becomes:
c
c
em =
from which r =
r
Substituting;
y
e x = - em
c
em
6
Stress Due to Bending
• In elastic region;
y
c
 x  E x   E m
y
   m (stress varies linearly)
c
• From static equilibrium,
y
Fx  0    x dA     m dA
c

0   m  y dA
c
• From static equilibrium,
 y

M    y x dA    y   m  dA
 c


 I
M  m  y 2 dA  m
c
c
m 
Mc M

I
S
y
Substituti ng  x    m
c
x  
My
I
7
Beam Section Properties
A=600 cm2
Mc M

I
S
I  section moment of inertia
m 
h=40 cm
h=30 cm
b=20 cm
b=15 cm
S
I
 section modulus
c
• For a rectangular cross section,
3
1
I 12 bh
S 
 16 bh3  16 Ah
c
h2
8
Example 4.01
20 mm
60 mm
A steel bar of 20x60 mm rectangular
cross section is subjected to two equal
and opposite couples acting in the
vertical plane of symmetry.
Determine the value of bending
moment M that causes the bar to
yield. Assume σy= 250 MPa.
9
Example 4.02
An aluminium rod with a semicircular
cross section of radius r=12 mm is bent
into the shape of a circular arc of mean
radius ρ=2.5 m. Knowing that the flat
face of the rod is turned toward the
center of curvature of the arc, determine
the maximum tensile and compressive
stress in the rod. Use E=70 MPa.
10
Beam Section Properties
11
Properties of American Standard Shapes
12
Deformations in a Transverse Cross Section
• Curvature of the neutral surface
1
r
=
em
c
1 M
=
r EI
=
sm
Ec
 y   x 
=
y

1 Mc
Ec I
 z   x 
y

1 
  anticlastic curvature
 
13
Sample Problem 4.1
120 mm
t=6 mm
80 mm
The rectangular tube shown is extruded from an aluminium alloy for
which σy= 275 MPa, σu= 415 MPa and E= 73 GPa. Neglecting the effect
of fillets, determine (a) the bending moment M for which the factor of
safety will be 3.00, (b) the corresponding radius of curvature of the tube.
14
Sample Problem 4.2
A cast-iron machine part is acted upon
by a 3 kN-m couple. Knowing E = 165
GPa and neglecting the effects of
fillets, determine (a) the maximum
tensile and compressive stresses, (b)
the radius of curvature.
15
Sample Problem 4.2
Area, mm 2
y , mm
yA, mm3
1 20  90  1800
50
90 103
2 40  30  1200
20
24 103
3
 A  3000
 yA  114 10
3
 yA 11410
Y 

 38 mm
3000
A

 
1 bh3  A d 2
I x   I  A d 2   12



1 90  203  1800122  1 30  403  1200 182
 12
12
I  868103 mm  86810-9 m 4
16

Sample Problem 4.2
Mc
I
M c A 3 kN  m  0.022 m
A 

I
868109 mm 4
M cB
3 kN  m  0.038 m
B  

I
868109 mm 4
m 
1



M
EI
1
3 kN  m
165 GPa 86810
-9
m
4

 A  76.0 MPa
 B  131.3 MPa
 20.95 103 m-1

  47.7 m
17
Bending of Members Made of Several Materials
y
x  

1  E1 x  
E1 y

 2  E2 x  
E2 y

Ey
E y
dF1  1dA   1 dA dF2   2dA   2 dA

x  
My
I
1   x
 2  n x
dF2  

nE1  y dA   E1 y n dA


E
n 2
E1
18
Example Problem 4.03
18 mm
10 mm
10 mm
75 mm
A bar obtained by bonding together pieces
of steel (Es= 200 GPa) and brass (Eb= 100
GPa)has the cross section shown.
Determine the maximum stress in the steel
and brass when the bar is in pure bending
with a bending moment M= 4.5 kN-m.
10 mm
36 mm
10 mm
37.5 mm
75 mm
Transformed cross-section:
56 mm
19
Sample Problem 4.3
Two steel plates have been welded together
to form a bema in the shape of a T that has
been strengthened by securely bolting it to
the two oak timbers shown. The modulus of
elasticity is 12.5 GPa for the wood and 200
Gpa for the steel. Knowing that a bending
moment M= 50 kN-m is applied to the
composite beam, determine (a) the
maximum stress in the wood, (b) the stress
in the steel along the top edge.
20
Reinforced Concrete Beams
• n = Es/Ec.
• Location of the neutral axis,
bx x  n As d  x   0
2
1 b x2
2
 n As x  n As d  0
• The normal stresses I concrte and steel;
x  
My
I
c   x
 s  n x
21
Sample Problem 4.4
100 mm
150 mm
140 mm
150 mm
150 mm
150 mm
A concrete floor slab is reinforced with 16-mm
diameter steel rods. The modulus of elasticity is
200 GPa for steel and 20GPa for concrete. With
an applied bending moment of 4.5 kN-m for 0.3 m
width of the slab, determine the maximum stress in
the concrete and steel.
22
Sample Problem 4.4
300 mm
100 mm
nAs = 3216 mm2
12.9 MPa
177.8 MPa
23
Stress Concentrations
Mc
m  K
I
24
Plastic Deformations
y
 x   m
c
and
My
x  
I
Fx    x dA  0
M    y x dA
25
Plastic Deformations
RB 
MU c
I
26
Members Made of an Elastoplastic Material
Mc
I
 x  Y
m 
 m  Y
I
M Y   Y  maximum elastic moment
c
2

y
M  32 M Y 1  13 Y2 

c 

yY  elastic core half - thickness
M p = 23 M Y = plastic moment (for rectangular X-sect)
Mp
k=
= shape factor (depends only on cross section shape)
MY
27
Members Made of an Elastoplastic Material
M ³ M y Þ yY = eY r
• Radius of curvature at onset of yield;
M = M y Þ c = eY rY
yY r
=
c rY
2ö
æ
r
M = 23 MY ç1- 13 2 ÷
rY ø
è
28
Members Made of an Elastoplastic Material
• For rectangular cross section;
2 2
M y = bc s y
3
M p = bc2s y
29
Plastic Deformations of Members with a
Single Plane of Symmetry
M p Zs y Z
k=
=
= (Z: Plastic section modulus)
MY Ss y S
R1  R2
A1 Y  A2 Y


M p  12 A Y d
30
Residual Stresses
31
Example Problem 4.05, 4.06
A member of uniform rectangular cross section is
subjected to a bending moment M = 36.8 kN-m.
The member is made of an elastoplastic material
with a yield strength of 240 MPa and a modulus
of elasticity of 200 GPa.
Determine (a) the thickness of the elastic core, (b)
the radius of curvature of the neutral surface.
After the loading has been reduced back to zero,
determine (c) the distribution of residual stresses,
(d) radius of curvature.
32
Example Problem 4.05, 4.06
• Thickness of elastic core:
M 
2

3 M 1  1 yY
2 Y
3 2


c 

36.8 kN  m 
2

3 28.8 kN  m 1  1 yY
2
 3 2

yY
yY

 0.666
c
60 mm


c 
2 yY  80 mm
• Radius of curvature:
Y 
• Maximum elastic moment:



2
I 2 2 2
 3 bc  3 50 103 m 60 103 m
c
 120 10 6 m3


I
M Y   Y  120 10 6 m3 240 MPa 
c
 28.8 kN  m
Y
E

240 106 Pa
200 109 Pa
 1.2 103
Y 
yY

yY

Y

40 103 m
1.2 103
  33.3 m
33
Example Problem 4.05, 4.06
• M = 36.8 kN-m
yY  40 mm
 Y  240 MPa
• M = -36.8 kN-m
Mc 36.8 kN  m

I
120 106 m3
 306.7 MPa  2 Y
 
m
• M=0
At the edge of the elastic core,
x 
x
E

 35.5 106 Pa
200 109 Pa
 177.5 10 6
 
yY
x

  225m
40 103 m
177.5 10 6
34
Sample Problem 4.5
25 mm
400 mm
Beam AB has been prefabricated from a
high-strength low-alloy steel that is
assumed to be elastoplastic with E= 200
GPa and σy= 350 MPa. Neglecting the
effect of fillets, determine the bending
moment M and the corresponding radius
of curvature (a) when yield first occurs,
(b) when flanges have just become fully
plastic.
25 mm
300 mm
35
Sample Problem 4.6
Determine the plastic moment Mp of a beam
with the cross section shown when the
beam is bent about a horizontal axis.
Assume that the material is elastoplastic
with a yield strength pf 240 MPa.
36
Eccentric Axial Loading in a Plane of Symmetry
 x   x centric   x bending

• Eccentric loading
P My

A I
FP
M  Pd
37
Example Problem 4.07
The largest allowable stresses for the cast
iron link are 30 MPa in tension and 120
MPa in compression. Determine the largest
force P which can be applied to the link.
700 N
12 mm
16 mm
700 N
38
Sample Problem 4.8
The largest allowable stresses for the cast
iron link are 30 MPa in tension and 120
MPa in compression. Determine the largest
force P which can be applied to the link.
• From sample problem 4.2;
A  3 103 m 2
Y  0.038 m
I  868109 m 4
39
Sample Problem 4.8
• Determine an equivalent centric and bending loads.
d  0.038  0.010  0.028 m
P  centric load
M  Pd  0.028 P  bending moment
• Superpose stresses due to centric and bending loads
0.028 P 0.022  377 P
P Mc A
P



A
I
3 103
868109
0.028 P 0.022  1559 P
P Mc
P
B    A  

A
I
3 103
868109
A  
• Evaluate critical loads for allowable stresses.
 A  377 P  30 MPa
P  79.6 kN
 B  1559 P  120 MPa P  79.6 kN
• The largest allowable load
P  77.0 kN
40
Unsymmetric Bending
41
Unsymmetric Bending
• 0  Fx    x dA     y  m dA
 c

or 0   y dA
neutral axis passes through centroid
 y 
• M  M z    y   m dA
 c

σ I
or M  m I  I z  moment of inertia
c
defines stress distribution
 y

0

M

z

dA

z



 x

y
m dA
•
 c

or 0   yz dA  I yz  product of inertia
Fx = 0 = M y
M z = M = applied couple
If couple vector is directed along a
principal centroidal axis, then neutral axis
will coincide with axis of the couple.42
Unsymmetric Bending
• Resolve the couple vector into components along
the principle centroidal axes.
M z  M cos
M y  M sin
• Superpose the component stress distributions
My M z
sx =- z + y
Iz
Iy
• Along the neutral axis,
M y M z ( M cosq ) y ( M sinq ) z
sx = 0 =- z + y =+
Iz
Iy
Iz
Iy
y I
tan f = = z tan q
z Iy
43
Example 4.08
180 N-m
90 mm
M z  M cos
M y  M sin
Mzy Myy
x  

Iz
Iy
A 180 N-m couple is applied to a
rectangular wooden beam in a plane
forming an angle of 30 deg. with the
vertical. Determine (a) the maximum stress
in the beam, (b) the angle that the neutral
axis forms with the horizontal plane.
tan  
y Iz
 tan 
z Iy
44
Example 4.08
-6.64 MPa
6.64 MPa
45
General Case of Eccentric Axial Loading
P  centric force
M y  Pa
M z  Pb
P Mz y M yz
x  

A
Iz
Iy
• Neutral axis may be found from
My
Mz
P
y
z
Iz
Iy
A
46
Example Problem 4.09
A vertical 4.80 kN load is applied as shown
on a wooden post of rectangular cross
section, 80 by 120 mm.
(a) Determine the stress at points A, B, C
and D.
(b) Locate the neutral axis of the cross
section.
47
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