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
write-up
on
elastic-plastic
beam bending
Consider the square cross-section 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/perfectly-plastic 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
highly-stressed
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 cross-sections 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
cross-section,
and
compressive
yielding
stress
values
of
magnitude
−
σ
y
in
the
other
triangular half of the cross-section.
At this point, the bending moment carried by the
cross-section 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 cross-section, but on rotated axes, so that the cross-section appears as a square?
(Our usual orientation for bending.)