tutorial presentation zhao

7/29/2019 Tutorial Presentation Zhao

1/42


2/42

Definition of normal stress(axial stress)

A

F


3/42

Definition of normal strain

0L

L


4/42

Poissons ratio


5/42

Definition of shear stress

0A

F


6/42

Definition of shear strain

l

x tan


7/42

Tensile Testing


8/42

Stress-Strain Curves


9/42

Stress-Strain Curves

http://www.uoregon.edu/~struct/courseware/461/461_lectures/4

61_lecture24/461_lecture24.html


10/42

Stress-Strain Curve

(ductile material)

http://www.shodor.org/~jingersoll/weave/tutorial/node4.html
http://www.shodor.org/~jingersoll/weave/tutorial/img21.png


11/42

Stress-Strain Curve

(brittle material)


12/42

Example: stress-strain curve for low-carbon steel

1 - Ultimate Strength

2 - Yield Strength

3 - Rupture

4 - Strain hardening region

5 - Necking region

Hooke's law is only valid for the

portion of the curve between the

origin and the yield point.

http://en.wikipedia.org/wiki/Hooke's_law
http://upload.wikimedia.org/wikipedia/commons/0/00/Stress_v_strain_A36_2.png


13/42

PLProportional Limit - Stress above which stress is not longer proportional to strain.

ELElastic Limit - The maximum stress that can be applied without resulting in permanent

deformation when unloaded.

YPYield Point - Stress at which there are large increases in strain with little or no increase in

stress. Among common structural materials, only steel exhibits this type of response.

YSYield Strength - The maximum stress that can be applied without exceeding a specified

value of permanent strain (typically .2% = .002 in/in).

OPTI 222 Mechanical Design in Optical Engineering 21

UUltimate Strength - The maximum stress the material can withstand (based on the original

area)


14/42

True stressand true strain

True stressand true strain are based uponinstantaneous values of cross sectional

area and gage length


15/42

The Region of Stress-Strain Curve

Stress Strain Curve

Similar to Pressure-Volume Curve

Area = Work

Volume

Pressure

Volume


16/42

Uni-axial Stress State

Elastic analysis


17/42

Stress-Strain Relationship

EE -- Youngs modulus

GG -- shear modulus

Hookes Law:


18/42

Stresses on Inclined Planes


19/42

Thermal Strain

Straincaused by temperature changes. is amaterial characteristic called the coefficient ofthermal expansion.


20/42

Strains caused by temperature changes and strainscaused by applied loads are essentially independent.

Therefore, the total amount of strain may be expressed as

follows:


21/42

Bi-axial stateelastic analysis


22/42

(1) Plane stress

State of plane stress occurs in a thin plate subjected to forces acting in the mid-plane of the

plate

State of plane stress also occurs on the free surface of a structural element or machine

component, i.e., at any point of the surface not subjected to an external force.


23/42

Transformation of Plane Stress


24/42

Mohrs Circle (Plane Stress)

http://www.tecgraf.puc-rio.br/etools/mohr/mohreng.html


25/42



26/42

Instruction to drawMohrs Circle1. Determine the point on the body in which the principal stresses are to be

determined.

2. Treating the load cases independently and calculated the stresses for the point

chosen.

3. Choose a set of x-y reference axes and draw a square element centered on the

axes.

4. Identify the stresses x, y, and xy = yx and list them with the proper sign.

5. Draw a set of - coordinate axes with being positive to the right and being

positive in the

upward direction. Choose an appropriate scale for the each axis.

6. Using the rules on the previous page, plot the stresses on the x face of the element

in this coordinate system (point V). Repeat the process for the y face (point H).

7. Draw a line between the two point V and H. The point where this line crosses the axis establishes the center of the circle.

8. Draw the complete circle.

9. The line from the center of the circle to point V identifies the x axis or reference

axis for angle measurements (i.e. = 0).

Note: The angle between the reference axis and the axis is equal to 2p.


27/42


http://www.egr.msu.edu/classes/me423/aloos/lecture_notes/lecture_4.pdf


28/42

Principal Stresses


29/42

Maximum shear stress


30/42

http://www4.eas.asu.edu/concrete/elasticity2_95/sld001.htm


(Plane stress)

))((1

yxzE

xy

y

x

xy

y

xE

2100

01

01

12


31/42

(2) Plane strain


32/42

Coordinate Transformation

The transformation of strains with respect to the {x,y,z} coordinates to

the strains with respect to {x',y',z'} is performed via the equations


33/42

Mohr's Circle (Plane Strain)

(xx' - avg)2

+ ( x'y' / 2 )2

= R2

avg =xx+ yy

2

http://www.shodor.org/~jingersoll/weave4/tutorial/tutorial.html
http://www.shodor.org/~jingersoll/weave4/tutorial/Figures/mc_strain.jpg


34/42

http://www.efunda.com/formulae/solid_mechani

cs/mat_mechanics/calc_principal_strain.cfm

Principal Strain


35/42

Maximum shear strain


36/42

Stress-Strain Relationship(Plane strain)

)(

211

yxz

E

z

y

x

z

y

xE

)1(2

2100

011

01

1

)21)(1(

)1(


37/42

Tri-axial stress state

elastic analysis


38/42

3D stress at a point

three (3) normal stresses may act on faces of the cube, as well

as, six (6) components of shear stress


39/42

Stress and strain components


40/42

The stress on a inclined plane

))(()2()2(3121

22322232

lnn

))(()2

()2

( 123222132213

mnn

))(()

2

()

2

( 231322212221

nnn

y

x

z

(l, m, n)

p2

3

1

n

n


41/42

3-D Mohrs Circle

* The 3 circles expressed by the 3 equations intersect in point D,

and the value of coordinates of D is the stresses of the inclined

plane

D


42/42


For isotropic materials

Generalized Hookes Law:

0

0

0

1

11

21

2

2100000

0

2

210000

002

21000

0001

0001

0001

)21)(1(

TEE

zx

yz

xy

z

y

x

zx

yz

xy

z

y

x

tutorial presentation zhao

Documents