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This lecture note is part of Material and Structures course. It was provided by Prof. Aparijita Singh at Andhra University. It includes: Plane, Stress, Configurations, Models, Stretching, Shearing, Slab, Displacement, Temperature, Stress, Magnitude, Equilibrium, Orthotropic, Isotropic
Typology: Exercises
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Figure 6.1 This deals with stretching and shearing of thin slabs.
Representation of Generic Thin Slab
The body has dimensions such that
h << a, b
Key
where are limits to “<<“???
We’ll
consider later)
displacement (and temperature) with respect to y Thus, the plate is thin enough such that there is no variation of
3 (z).
Figure 6.2 magnitude). Furthermore, stresses in the z-direction are zero (small order of
Representation of Cross-Section of Thin Slab
Stress-Strain
(fully anisotropic)
Primary
(in-plane) strains 1
− ν
12 σ 2^
− η
16 (^) σ 6
ε 2^
=
2
− ν
21
σ 1
σ 2^
− η
26 (^) σ 6
−η
61
σ 1
− η
62 σ 2^
σ 6
Invert
to get:
σ αβ
αβσγ
ε σγ
Secondary
(out-of-plane) strains
(they exist, but they are not a
primary
part of the problem)
− ν
31 σ 1
− ν
32
σ 2
− η
36
σ 6
− η
41
σ 1
− η
42
σ 2
− η
46
σ 6
−η
51
σ 1
− η
52
σ 2
− η
56
σ 6
Note
: can reduce these for orthotropic, isotropic
(etc.) as before.
Strain - Displacement
Primary
ε 11
=
u 1
y 1
ε 22
=
u 2
y 2
ε 12
=
∂ u 1 + ∂ u 2
y 2
y 1 (^)
This further implies from above
(since
y 3
No in-plane variation
u 3
y α
but this is not exactly true
are, in actuality, triaxial ( Why? This is an idealized model and thus an approximation. There
σ zz , etc.) stresses that we ignore here as
being
small
relative to the in-plane stresses!
(we will return to try to define “small”)
Final note
: for an orthotropic material, write the tensorial
stress-strain equation as:
2-D plane stress
σ
αβ
ε σγ
(
,
,
α β , σ , γ =
1 2)
αβσγ ∗
There is
not
a 1-to-1 correspondence between the 3-D E
mnpq
and
the 2-D E
αβσγ
. The effect of
ε 33
must be incorporated since
ε 33
does
not appear in these equations by using the (
σ 33
= 0) equation.
This gives:
ε 33
= f(
ε αβ )
the case of plane stress in place of engineering notation: Also, particularly in composites, another “notation” will be used in
subscript
x = 1 = L (longitudinal)…along major axis
change
y = 2 = T (transverse)…along minor axis
This deals with long prismatic bodies:
Key
again: where are limits to “>>”??? … we’ll
consider later)
x - y plane (none in z) and do Since the body is basically “infinite” along z, the important loads are in the
not
change with z:
y 3
z
rigid body movement): This implies there is no gradient in displacement along z, so (excluding
u 3 = w = 0
Equations of elasticity become:
Equilibrium:
Primary
∂σ
11
∂σ
21
1 f
=
∂ y 1 ∂ y 2 ∂σ
12
∂σ
22
f 2
=
∂ y 1 ∂ y 2
Unit 6 - p. 11
Secondary
∂σ
13
∂σ
23
f 3
=
∂ y 1 ∂ y 2 σ
13
and
σ 23
exist but do not enter into
primary
consideration
Strain - Displacement
ε 11
=
u 1
y 1
ε 22
=
u 2
y 2
ε 12
=
∂ u 1 + ∂ u 2
y 2
y 1 (^)
Assumptions
w
give:
y 3
ε 13
ε 23
ε 33
Plane strain
Plane Stress
Eliminate Plane Strain
σ 33
from eq.
Set by using
σ 33
σ
ε
eq. and expressing
σ 33
in terms of
ε αβ
Eliminate
ε 33
from eq. set
by using
σ 33
= 0
σ
ε eq.
and expressing
ε 33
in
terms of
ε αβ
Note:
σ 33
ε 33,
u 3
Variable(s):Secondary
ε αβ
, σ αβ , u
α
ε αβ
, σ αβ , u
α
Variables:Primary
ε i
= 0
σ i = 0
Assumptions:Resulting
σ αβ
only
y 3 = 0
σ 33
<<
σ αβ
Loading:
length (y
3 ) >> in-plane
dimensions (y
1 , y
2 (^) )
thickness (y
3 ) << in-plane
dimensions (y
1 , y
2 )
Geometry:
Figure 6.4 Plane Stress:
Pressure vessel (fuselage, space habitat) Skin
(10 ksi) in order of 70 MPa
p o ≈
70 kPa (~ 10 psi for living environment)
⇒ σ
zz
<<
σ
xx
, σ
yy
, σ
xy
Fall, 2002
but…when do these apply???
Depends on
loading
geometry
material and its response
issues of scale
how “good” do I need the answer
what are we looking for (deflection, failure, etc.)
We’ve talked about the first two, let’s look a little at each of the last three:
--> Material and its response
magnitude of “primary” / “secondary” factorsElastic response and coupling changes importance /
Key
: are “primary” dominating the response?)
--> Issues of scale
What am I using the answer for? at what level?
Example
--overall deflection or reactions in legs are not: standing on table dependent on way I stand (tip toe or flat foot)
model of top of table as plate in
bending is sufficient
--stresses under my foot
very
sensitive to
specifics
(if table top is foam, the way I stand
will determine whether or not I
crush the foam)
--> How “good” do I need the answer?
design, need “exact” numbersIn preliminary design, need “ballpark” estimate; in final
longer in plane stressExample: as thickness increases when is a plate no
