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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
CAMBRIDGE, MASSACHUSETTS 02139
2.002 MECHANICS AND MATERIALS II
SOLUTIONS
FOR
HOMEWORK
NO.
4
Problem 1
(20
points)
Part
A:
Because
the
beam
is
still
in
elastic
region,
the
stress
field
can
be
expressed
as:
σ
(
y
) =
−
yM
y
I
(1)
Since
the
most
highly
stressed
region
is
at
the
verge
of
yielding,
we
have

σ
(
y
)
max

=
 −
yM
y
I

y
=
±
h/
√
2
=
σ
y
⇒
M
y
=
σ
y
I
h/
√
2
(2)
For
the
diamondorientation
beam,
I
is
calculated
as:
h
√
h
4
h
2
2
dA
= 2
(
√
−
y
)
×
2
dy
=
I
(3)
=
y
12
2
0
The
area
moment
of
inertia
for
this
diamondorientation
is
the
same
as
the
crosssection
appears
as
a
square.
Substitution
of
I
into
Eq.
2,
we
get
that:
√
M
2
σ
y
h
3
y
(
diamond
)
=
(4)
12
Part
B:
√
h
√
h
3
h
2
σ
2
y
M
L
(
diamond
)
=
σ
y
(
−
y
)
dA
= 2
σ
y
y
×
(
√
−
y
)
×
2
dy
=
(5)
6
2
0
Part
C:
√
M
L
(
diamond
)
2
σ
y
h
3
6
=
√
= 2
(6)
2
σ
y
h
3
M
y
(
diamond
)
12
σ
When
the
crosssection
appears
as
a
square,
M
y
is
calculated
as:
y
I
σ
y
h
4
/
12
σ
y
h
3
M
=
=
=
y
(
square
)
y
max
h/
2
6
1
(7)
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and
M
L
is:
h
M
h
3
L
(
square
)
=
σ
(
−
y
)
dA
= 2
σ
y
y
×
hdy
=
(8)
2
σ
y
4
0
So
the
ratio
of
M
L
/M
y
for
squareorientation
is:
M
L
(
square
)
σ
y
h
3
3
4
M
=
=
(9)
y
(
square
)
σ
y
h
3
2
6
Part
D:
The
ratio
of
the
firstyield
bending
moments
for
the
two
orientations
is:
√
M
y
(
diamond
)
2
σ
y
h
3
√
12
M
=
= 1
/
2
(10)
y
(
square
)
σ
y
h
3
6
The
firstyield
bending
moment
is
calculated
by:
σ
y
I
y
M
y
=
(11)
max
Since
σ
y
is
a
constant,
and
I
is
the
same
for
these
two
orientations,
the
above
ratio
is
only
determined
by
the
ratio
of
y
max
.
The
ratio
of
limit
moment
for
the
two
orientations
is:
√
M
L
(
diamond
)
2
σ
y
h
3
2
√
6
M
=
=
2
<
1
.
0
(12)
L
(
square
)
σ
y
h
3
3
4
The
limit
moment
for
a
symmetrical
crosssection
is
calculated
by:
M
L
= 2
σ
y
ydA
(13)
A/
2
where
A/
2 is the
1
/
2
area
of
the
cross
section
in
which
y
≥
0.
For
the
diamondorientation,
√
though
the
maximum
value
of
y
is
2
times
larger
than
the
squareorientation,
the
major
part
of
its
crosssection
is
located
at
the
region
with
small
y
coordinate.
Thus
the
ratio
of
limit
moment
calculated
above
is
smaller
than
one.
Since
M
y
(
diamond
)
< M
y
(
square
)
and
M
L
(
diamond
)
< M
L
(
square
)
,
we
should
use
the
beam
in
the
squareorientation.
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 Spring '04
 DavidParks
 Stress, Strength of materials, dislocation motion

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