Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Plane Stress-Material and Structures-Lecture Handout, Exercises of Structures and Materials

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

2011/2012

Uploaded on 07/26/2012

raam
raam 🇮🇳

4.4

(14)

90 documents

1 / 18

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
There are many structural configurations where we do not
have to deal with the full 3-D case.
First let’s consider the models
Let’s then see under what conditions we can
apply them
A. Plane Stress
This deals with stretching and shearing of thin slabs.
Figure 6.1 Representation of Generic Thin Slab
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12

Partial preview of the text

Download Plane Stress-Material and Structures-Lecture Handout and more Exercises Structures and Materials in PDF only on Docsity!

have to deal with the full 3-D case.There are many structural configurations where we do not

First let’s consider the models

apply themLet’s then see under what conditions we can

A. Plane Stress

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

ε 1 = E 1 [ σ 1

− ν

12 σ 2^

− η

16 (^) σ 6

]

ε 2^

=

E

2

[

− ν

21

σ 1

σ 2^

− η

26 (^) σ 6

]

ε 6 = G 6 [

−η

61

σ 1

− η

62 σ 2^

σ 6

]

Invert

to get:

σ αβ

E

αβσγ

ε σγ

Secondary

(out-of-plane) strains

(they exist, but they are not a

primary

part of the problem)

ε 3 = E 3 [

− ν

31 σ 1

− ν

32

σ 2

− η

36

σ 6

]

ε 4 = G 4 [

− η

41

σ 1

− η

42

σ 2

− η

46

σ 6

]

ε 5 = G 5 [

−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

INCONSISTENCY

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 ∗

E

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

The other important “extreme” model is…

B. Plane Strain

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

SUMMARY

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:

Examples

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