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8/7/2019 unit6 mit plane stress and strain
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MIT - 16.20 Fall, 2002
Unit 6Plane Stress and Plane Strain
Readings:
T & G 8, 9, 10, 11, 12, 14, 15, 16
Paul A. Lagace, Ph.D.Professor of Aeronautics & Astronautics
and Engineering Systems
Paul A. Lagace 2001
8/7/2019 unit6 mit plane stress and strain
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There are many structural configurations where we do not
have to deal with the full 3-D case. First lets consider the models
Lets then see under what conditions we canapply them
A. Plane Stress
This deals with stretching and shearing of thin slabs.
Figure 6.1 Representation of Generic Thin Slab
Paul A. Lagace 2001 Unit 6 - p. 2
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The body has dimensions such thath
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Stress-Strain (fully anisotropic)
Primary (in-plane) strains1
1 =E1
[1 122 166] (3)1
2 = E2 [ 21 1 + 2 266] (4)1
6 =G6
[61 1 622 + 6] (5)Invert to get:
* = E Secondary (out-of-plane) strains (they exist, but they are not a primary part of the problem)
13 =
E3[ 311 322 366 ]
Paul A. Lagace 2001 Unit 6 - p. 5
8/7/2019 unit6 mit plane stress and strain
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14 =
G4
[ 41 1 42 2 466 ]1
5 =G5
[51 1 522 566 ]Note: can reduce these for orthotropic, isotropic
(etc.) as before.
Strain - Displacement
Primary
11 = u1 (6)y122 = u2 (7)
y2
12 = 1 u1 + u2 (8)2 y2 y1
Paul A. Lagace 2001 Unit 6 - p. 6
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Secondary
13 = 1 u1 + u3 2 y3 y1
23 = 1 u2 + u3 2 y3 y2
33 = u3y3Note: that for an orthotropic material
(23 ) (13)4 = 5 = 0 (due to stress-strain relations)
Paul A. Lagace 2001 Unit 6 - p. 7
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This further implies from above
(since = 0 )y3No in-plane variation
u3= 0
ybut this is not exactly true
INCONSISTENCYWhy? This is an idealized model and thus an approximation. There
are, in actuality, triaxial (zz, etc.) stresses that we ignore here asbeing 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
= ( , , , , = 1 2)E
Paul A. Lagace 2001 Unit 6 - p. 8
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Figure 6.3 Representation of Long Prismatic Body
Dimension in z - direction is much, much larger than inthe x and y directions
L >> x, yPaul A. Lagace 2001 Unit 6 - p. 10
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(Key again: where are limits to >>??? wellconsider later)
Since the body is basically infinite along z, the important loads are in thex - y plane (none in z) and do not change with z:
= = 0
y3 z
This implies there is no gradient in displacement along z, so (excludingrigid body movement):
u3 = w = 0
Equations of elasticity become:
Equilibrium:
Primary11
+ 21 + f1 = 0 (1)y1 y2
12+
22+ f2 = 0 (2)y1 y2
Paul A. Lagace 2001 Unit 6 - p. 11
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Secondary13
+23
+ f3 = 0y1 y213 and 23 exist but do not enter into primary
consideration
Strain - Displacement
11 = u1 (3)y1
22= u2
(4)y2
12 = 1 u1 + u2 (5)2 y2 y1
Assumptions = 0, w = 0 give: y3
13 = 23 = 33 = 0(Plane strain)
Paul A. Lagace 2001 Unit 6 - p. 12
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Stress - Strain
(Do a similar procedure as in plane stress)
3 Primary
11 = ... (6)22 = ... (7)12 = ... (8)
Secondary
13 = 023 = 0
orthotropic ( 0 for anisotropic)
33 0INCONSISTENCY: No load along z,
yet 33 (zz) is non zero.
Why? Once again, this is an idealization. Triaxial strains (33)actually arise.
You eliminate 33 from the equation set by expressing it in terms of
via (33) stress-strain equation.Paul A. Lagace 2001 Unit 6 - p. 13
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SUMMARY Plane Stress Plane Strain
Eliminate 33 from eq.Set by using 33 - eq. and expressing 33in terms of
Eliminate 33 from eq. setby using 33 = 0 - eq.and expressing 33 interms of
Note:
3333, u3SecondaryVariable(s):
, , u, , uPrimaryVariables:
i3 = 0i3 = 0ResultingAssumptions:
only/y3 = 0
33 > in-planedimensions (y1, y2)
thickness (y3)
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Examples
Plane Stress:
Figure 6.4 Pressure vessel (fuselage, space habitat) Skin
in order of 70 MPa(10 ksi)
po 70 kPa (~ 10 psi for living environment) zz
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Plane Strain:
Figure 6.5
Dams
waterpressure
Figure 6.6 Solid Propellant Rockets
high internal pressurePaul A. Lagace 2001 Unit 6 - p. 16
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butwhen 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.)
Weve talked about the first two, lets look a little at each of the last three:
--> Material and its response
Elastic response and coupling changes importance /magnitude of primary / secondary factors
(Key: are primary dominating the response?)
--> Issues of scale What am I using the answer for? at what level?
Example: standing on table
--overall deflection or reactions in legs are not
dependent on way I stand (tip toe or flat foot)Paul A. Lagace 2001 Unit 6 - p. 17
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model of top of table as plate inbending is sufficient
--stresses under my foot very sensitive tospecifics
(if table top is foam, the way I standwill determine whether or not I
crush the foam)
--> How good do I need the answer?
In preliminary design, need ballpark estimate; in finaldesign, need exact numbers
Example: as thickness increases when is a plate nolonger in plane stress
Paul A. Lagace 2001 Unit 6 - p. 18
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Figure 6.7 Representation of the continuum from plane stress toplane strain
very thick
a continuumvery thin
(plane stress)
(plane strain)
No line(s) of demarkation. Numbers
approach idealizations but never getto it.
Must use engineering judgment
AND
Clearly identify key assumptions in model and resulting limitations
Paul A. Lagace 2001 Unit 6 - p. 19