unit6 mit plane stress and strain

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


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MIT - 16.20 Fall, 2002

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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MIT - 16.20 Fall, 2002

The body has dimensions such thath


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MIT - 16.20 Fall, 2002

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 ]



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MIT - 16.20 Fall, 2002

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



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MIT - 16.20 Fall, 2002

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)



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MIT - 16.20 Fall, 2002

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



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MIT - 16.20 Fall, 2002

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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MIT - 16.20 Fall, 2002

(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



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MIT - 16.20 Fall, 2002

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)



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MIT - 16.20 Fall, 2002

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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MIT - 16.20 Fall, 2002

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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MIT - 16.20 Fall, 2002

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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MIT - 16.20 Fall, 2002

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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MIT - 16.20 Fall, 2002

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



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MIT - 16.20 Fall, 2002

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


unit6 mit plane stress and strain

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