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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
CAMBRIDGE, MASSACHUSETTS 02139
2.002 MECHANICS AND MATERIALS II
HOMEWORK NO. 4
Distributed
:
Friday, April 2, 2004
Due
:
Friday, April 9, 2004
Problem 1
(20 points)
Note:
for
reference
material,
consult
the
laboratory
writeup
on
elasticplastic
beam bending
Consider the square crosssection beam shown, of dimensions
h
by
h
, subject to “diamond
orientation”
bending
in
the
plane
shown
(neutral
axis:
plane
y
=
0).
The
beam
can
be
considered to be composed of an elastic/perfectlyplastic material having Young’s modulus
E
, and tensile yield strength
σ
y
.
1. Using the standard assumptions of engineering beam theory, evaluate the magnitude
of
applied
moment,
M
y
,
just
suﬃcient
to
bring
the
most
highlystressed
region
to
the verge of yielding.
Express your answer in terms of
h
and material properties, as
appropriate.
(Aside:
are you “surprised” by the value you got for
I
=
y
2
dA
in this
orientation?)
2. If the applied curvature is increased to very large values,
the elastic/plastic bound
aries
(tension
and
compression
sides)
in
this
geometry,
like
those
in
the
bending
of
rectangular crosssections studied earlier, will move inward, toward the neutral axis.
At “inﬁnite” curvature, the boundaries will reach opposite sides of the
y
= 0 surface,
resulting in tensile yielding stress values of magnitude
σ
y
in one “triangle” half of the
crosssection,
and
compressive
yielding
stress
values
of
magnitude
−
σ
y
in
the
other
triangular half of the crosssection.
At this point, the bending moment carried by the
crosssection reaches a limiting value,
M
L
.
Evaluate
M
L
for this section.
3. Using your answers to the two previous questions, evaluate the ratio
M
L
/M
y
for bend
ing of this section.
How does this value compare with the ratio for bending of this
same crosssection, but on rotated axes, so that the crosssection appears as a square?
(Our usual orientation for bending.)
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 Spring '04
 DavidParks
 Mechanical Engineering

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