Sample Problem 4.2 - AML140

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© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

1


Sample Problem 4.2

SOLUTION:


Based on the cross section geometry,
calculate the location of the section
centroid and moment of inertia.










2
d
A
I
I
A
A
y
Y
x

Apply the elastic flexural formula to
find the maximum tensile and
compressive stresses.

I
Mc
m



Calculate the curvature

EI
M


1
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.

© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

2


Sample Problem 4.2

SOLUTION:

Based on the cross section geometry, calculate
the location of the section centroid and
moment of inertia.

mm

38
3000
10
114
3






A
A
y
Y











3
3
3
3
2
10
114
3000
10
4
2
20
1200
30
40
2
10
90
50
1800
90
20
1
mm

,
mm

,
mm

Area,
A
y
A
A
y
y








4
9
-
4
3
2
3
12
1
2
3
12
1
2
3
12
1
2
m
10
868


mm
10
868
18
1200
40
30
12
1800
20
90



















I
d
A
bh
d
A
I
I
x
© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

3


Sample Problem 4.2


Apply the elastic flexural formula to find the
maximum tensile and compressive stresses.

4
9
4
9
m
10
868
m
038
.
0
m
kN

3
m
10
868
m
022
.
0
m
kN

3















I
c
M
I
c
M
I
Mc
B
B
A
A
m



MPa

0
.
76


A

MPa

3
.
131


B


Calculate the curvature





4
9
-
m
10
868
GPa

165
m
kN

3
1




EI
M

m

7
.
47
m
10
95
.
20
1
1
-
3






© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

4


Bending of Members Made of Several Materials


Normal strain varies linearly.



y
x



Piecewise linear normal stress variation.







y
E
E
y
E
E
x
x
2
2
2
1
1
1






Neutral axis does not pass through
section centroid of composite section.


Elemental forces on the section are

dA
y
E
dA
dF
dA
y
E
dA
dF




2
2
2
1
1
1







Consider a composite beam formed from
two materials with
E
1

and
E
2
.

x
x
x
n
I
My









2
1




1
2
1
1
2
E
E
n
dA
n
y
E
dA
y
nE
dF








Define a transformed section such that

© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

5


Example 4.03

SOLUTION:


Transform the bar to an equivalent cross
section made entirely of brass


Evaluate the cross sectional properties of
the transformed section


Calculate the maximum stress in the
transformed section. This is the correct
maximum stress for the brass pieces of
the bar.


Determine the maximum stress in the
steel portion of the bar by multiplying
the maximum stress for the transformed
section by the ratio of the moduli of
elasticity.

Bar is made from bonded pieces of
steel (
E
s

= 200 GPa) and brass (
E
b

= 100 GPa). Determine the
maximum stress in the steel and
brass when a moment of 4.5 KNm
is applied.

© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

6


Example 4.03


Evaluate the transformed cross sectional properties





4
6
3
12
1
3
12
1
mm

10
96875
.
1
mm

75
mm

56




h
b
I
T
SOLUTION:


Transform the bar to an equivalent cross section
made entirely of brass.

mm

56
mm

10
mm

18
2
mm

10
0
.
2
GPa
100
GPa
200








T
b
s
b
E
E
n

Calculate the maximum stresses





MPa

7
.
85
m

10
1.96875
m

.0375
0
Nm

4500
4
6
-




I
Mc
m





MPa

7
.
85
2
max
max




m
s
m
b
n








MPa

.4
71
1
MPa

7
.
85
max
max


s
b


© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

7


Reinforced Concrete Beams


Concrete beams subjected to bending moments are
reinforced by steel rods.


To determine the location of the neutral axis,





0
0
2
2
2
1






d
A
n
x
A
n
x
b
x
d
A
n
x
bx
s
s
s

The steel rods carry the entire tensile load below
the neutral surface. The upper part of the
concrete beam carries the compressive load.


In the transformed section, the cross sectional area
of the steel,
A
s
,
is replaced by the equivalent area

nA
s

where
n = E
s
/E
c
.


The normal stress in the concrete and steel

x
s
x
c
x
n
I
My









© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

8


Sample Problem 4.4

SOLUTION:


Transform to a section made entirely
of concrete.


Evaluate geometric properties of
transformed section.


Calculate the maximum stresses
in the concrete and steel.

A concrete floor slab is reinforced with 16
-
mm
-
diameter steel rods. The modulus of
elasticity is 200 GPa for steel and 25 GPa
for concrete. With an applied bending
moment of 4.5 kNm for 0.3 m width of the
slab, determine the maximum stress in the
concrete and steel.

© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

9


Sample Problem 4.4


Evaluate the geometric properties of the
transformed section.











4
6
2
2
3
3
1
mm
10
8
.
12
mm
2
.
63
mm
3216
mm
8
.
36
mm
300
mm
8
.
36
0
100
3216
2
300














I
x
x
x
x
SOLUTION:


Transform to a section made entirely of concrete.





2
2
4
mm
3216
mm
16
2
0
.
8
0
.
8
GPa

5
2
GPa

00
2







s
c
s
nA
E
E
n

Calculate the maximum stresses.

4
6
-
2
4
6
-
1
m
10
8
.
2
1
m
0632
.
0
Nm
4500
0
.
8
m
10
8
.
2
1
m
0368
.
0
Nm
4500








I
Mc
n
I
Mc
s
c


MPa
9
.
12

c

MPa
8
.
177

s

© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

10


Stress Concentrations

Stress concentrations may occur:


in the vicinity of points where the
loads are applied

I
Mc
K
m



in the vicinity of abrupt changes
in cross section

© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

11


Plastic Deformations


For any member subjected to pure bending

m
x
c
y




strain varies linearly across the section


If

the member is made of a
linearly elastic material
,
the neutral axis passes through the section centroid

I
My
x



and


For a member with vertical and horizontal planes of
symmetry and a material with the same tensile and
compressive stress
-
strain relationship, the neutral
axis is located at the section centroid and the stress
-
strain relationship may be used to map the strain
distribution from the stress distribution.


For a material with a nonlinear stress
-
strain curve,
the neutral axis location is found by satisfying







dA
y
M
dA
F
x
x
x


0
© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

12


Plastic Deformations


When the maximum stress is equal to the ultimate
strength of the material, failure occurs and the
corresponding moment
M
U

is referred to as the
ultimate bending moment
.


R
B

may be used to determine
M
U

of any
member made of the same material and with the
same cross sectional shape but different
dimensions.


The
modulus of rupture in bending, R
B
, is found
from an experimentally determined value of
M
U

and a fictitious linear stress distribution.

I
c
M
R
U
B

© 2009 The McGraw
-
Hill Companies, Inc. All rights reserved.

MECHANICS OF MATERIALS

Fifth

Edition

Beer


Johnston


DeWolf


Mazurek


4
-

13


Members Made of an Elastoplastic Material


Rectangular beam made of an elastoplastic material

moment

elastic

maximum






Y
Y
Y
m
m
Y
x
c
I
M
I
Mc







If the moment is increased beyond the maximum
elastic moment, plastic zones develop around an
elastic core.

thickness
-
half

core

elastic

1
2
2
3
1
2
3











Y
Y
Y
y
c
y
M
M

In the limit as the moment is increased further, the
elastic core thickness goes to zero, corresponding to a
fully plastic deformation.

shape)
section

cross
on
only

(depends
factor

shape

moment

plastic

2
3




Y
p
Y
p
M
M
k
M
M