StressPresentation.ppt

8/14/2019 StressPresentation.ppt

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B. NOWAK, StressPrinciples., CASA Seminar, 8thMarch 2006

Stress

and

Stress Princ iples

CASA Seminar


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Basic Overview

Stress Definitions

Cauchy Stress Principle

The Stress Tensor

Principal Stresses, Principal Stress Direction

Normal and Shear Stress Components

Mohrs Circles for Stress

Special Kinds of Stress

Numerical Examples of Stress Analysis



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Stress definitions

Stress a measure of force

intensity, either within or on the

bounding surface of a body

subjected to loads

Stress - a medical term for a wide

range of strong external stimuli,

both physiological and

psychological



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Stress(Basic assumptions and definitions)

In continuum mechanics a body is consideredstress freeif the only forces present are those inter-atomic forces required to hold the body together

Basic types of forces are distinguishedfrom one another:

Body forces i.e.: gravity, inertia; designated by vectorsymbol bi (force per unit mass)

or pi (force per unit volume); acting on all volumeelements, and distributed throughout the body;

Surface forces i.e.: pressure; denoted by vector symbolfi (force per unit area of surface across they which theyact); act upon and are distributed in some fashion overa surface element of the body, regardless of whetherthat element is part of the bounding surface, or anarbitrary element of surface within the body;

External forces acting on a body (loads applied to thebody);

Internal forces acting between two parts of the body(forces which resist the tendency for one part of themember to be pulled away from another part).


bipi

fi


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Stress(Density definition)

In continuum mechanics we consider a

material body B having a volume V enclosed by a

surface S, and occupying a regular region R0 of

physical space. Let P be an interior point of the

body located in the small element of volume V

whose mass is M. We define the average density

of this volume element by the ratio:

V

mave


and the density at point P by the limit of this ratio

as the volume shrinks to the point:

dV

dm

V

m

V

0lim

V

mave

The units of density are kilograms per cubic meter

(kg/m3). Two measures of body forces, bihaving unitsof Newtons per kilogram (N/kg), andpihaving units of

Newtons per meter cubed (N/m3), are related through

the density by the equation:

ii pb pb

The density is in general a scalar function ofposition and time:

),( tx


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Cauchy Stress Principle

)(

**0*

lim n

iii

St

dS

df

S

f


We consider a homogeneous, isotropic material body B having a bounding surface S, and a volume V, which is

subjected to arbitrary surface forces fiand body forces bi. Let P be an interior point of B and imagine a plane surface S* passing

through point P (sometimes referred to as a cutting plane) so as to partition the body into two portions, designated I and II.

Point P is in the small element of area S*of the cutting plane, which is defined by the unit normal pointing in the

direction from Portion I into Portion II as shown by the free body diagram of Portion I The internal forces being transmitted

across the cutting plane due to the action of Portion II upon Portion I will give rise to a force distribution on S*equivalentto a

resultant force fiand a resultant moment Miat P.

The Cauchy stress principle asserts that in the limit as the area S*shrinks to zero with P remaining an interior

point, we obtain:

0lim*

0*

S

MiS

fi


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The Stress Tensor(rectangular Cartesian components)


We can introduce a rectangular Cartesian reference frame at P, there is associated with each of the areaelements dSi(i= 1,2,3) located in the coordinate planes and having unit normals (i = 1,2,3), respectively, a stress vector as

shown in figure. In terms of their coordinate components these three stress vectors associated with the coordinate planes are

expressed by:

3

)(

32

)(

21

)(

1

)(

3

)(

32

)(

21

)(

1

)(

3

)(

32

)(

21

)(

1

)(

3333

2222

1111

eeet

eeet

eeet

eeee

eeee

eeee

ttt

ttt

ttt

This equation expresses the stress vector at P point for a given coordinate plan in terms of its rectangular Cartesian

components.

or using summation convention:

jjii t et ee )

()( )3,2,1( i


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The Stress Tensor(analysis for arbitrary oriented plane)


For this purpose, we consider the equilibrium of a small portion of the body in the shape ofa tetrahedron having its vertex at P, and its base ABC perpendicular to an arbitrarily oriented normal

The stress vectors shown on the surfaces of the tetrahedron

represent average values over the areas on which they act. This is indicated in

our notation by an asterisk appended to the stress vector symbols (remember

that the stress vector is a point quantity). Equilibrium requires the vector sum of

all forces acting on the tetrahedron to be zero, that is, for,

iin en

0***** 3)

(2)

(1)

()

( 321 dVbdStdStdStdSt iiiii

eeen

Now, taking into consideration area surfaces, volume we can rewrite above:

0*** )()( dVbdStdSt iii

j en

Now, letting the tetrahedron shrink to point P we get :

jii ntt j )()( en

0*3

1**

)()( hbntt ijiij

en

or by defining

)( jiji t e

nt n

n

)(

)(

jjii nt


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The Stress Tensor


The Cauchy stress formula expresses the stress vector associated with the element of area having an outward

normal niat point P in terms of the stress tensor components jiat that point. And although the state of stress at P has been

described as the totality of pairs of the associated normal and traction vectors at that point, we see from the analysis of the

tetrahedron element that if we know the stress vectors on the three coordinate planes of any Cartesian system at P, orequivalently, the nine stress tensor components jiat that point, we can determine the stress vector for any plane at that

point. For computational purposes it is often convenient to express it in the matrix form:

333231

232221

131211

321

)(

3

)(

2

)(

1 ,,,,

nnnttt nnn

Note that, for these, the first subscript designates the coordinate plane on which the shear stress acts, and the second

subscript identifies the coordinate direction in which it acts. A stress component is positive when its vector arrow points in the

positive direction of one of the coordinate axes while acting on a plane whose outward normal also points in a positive

coordinate direction. In general, positive normal stresses are called tensile stresses, and negative normal stresses are

referred to as compressive stresses. The units of stress are Newtons per square meter (N/m2) in the SI system One Newton

per square meter is called a Pascal, but because this is a rather small stress from an engineering point of view, stresses are

usually expressed as mega-Pascals (MPa).

The nine components of are often displayed by arrows on the coordinate faces of a

rectangular parallelepiped, as shown in figure. In an actual physical body B, all nine

stress components act at the single point P. The three stress components shown by

arrows acting perpendicular (normal) to there respective coordinate planes and labeled

11, 22, and 33 are called normal stresses. The six arrows lying in the coordinate

planes and pointing in the directions of the coordinate axes, namely, 12, 21, 23, 32,

31, and 13are called shear stresses.


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Principal Stresses, Principal Stress Directions

1iinn


The determination of principal stress values and principal stress directions follows precisely the procedure for

determining principal values and principal directions of any symmetric second-order tensor. In properly formulating the

eigenvalue problem for the stress tensor we use the identity:

and the substitution property of the Kronecker delta allows to rewrite:

nt n

n

)(

)(

jjii nt

0 jijji nIn the three linear homogeneous equations expressed implicitly above, the tensor components ijare assumed known; the

unknowns are the three components of the principal normal ni, and the corresponding principal stress . To complete thesystem of equations for these four unknowns, we use the normalizing condition on the direction cosines,

For non-trivial solutions the determinant of coefficients on njmust vanish. That is,

0 ijji

which upon expansion yields a cubic in (called the characteristic equation of the stress tensor)

023 IIIIII

whose roots (1), (2), (3) are the principal stress values of ij. The coefficients are known as

the first, second, and third invariants, respectively, of ijand may be expressed in terms of its components by:

O S C S S 8th


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Principal Stresses, Principal Stress Direction


det

2

1

2

1

321

22

kjiijk

jiijjjii

ii

III

trtrII

trI

Because the stress tensor ijis asymmetric tensor having real components, the three stress invariants are real, and

likewise, the principal stresses being roots are also real.

Directions designated by nifor which above equation is valid are calledprincipal stress directions, and the scalar is called

aprincipal stress value of ij. Also, the plane at P perpendicular to niis referred to as aprincipal stress plane. We see from

figure that because of the perpendicularity of to the principal planes, there are no shear stresses acting in these planes.

ii nt )(n

)(nt

B NOWAK St P i i l CASA S i 8th M h 2006


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Normal and Shear Stress Components

iiN nt )

(n


Also, from the geometry of decomposition, we get:

jiji nt )(n

nt )n(

N

ijijN

or

nn

The stress vector on an arbitrary plane at P may be resolved into a component normal to the plane having a magnitude N,

along with a shear component which acts in the plane and has a magnitude S, as shown in figure. (Here, Nand Sare notvectors, but scalar magnitudes of vector components. The subscripts N and S are to be taken as part of the component

symbols.) Clearly, from this figure, it is seen that Nis given by the dot product, and in as much as , it

follows that:

2

2)()(

NS

NiiS tt

)n()n(

nn

tt

B NOWAK St P i i l CASA S i 8th M h 2006


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Stress Matrix Components

T* AA

or

aa qmjmiqij *


Lets consider:

7500

0500

0025

5

30

5

40105

40

5

3

43024

0500

24057

5

30

5

40105

40

5

3

*

ij

Shear stress components

Normal stress components Principal stress components

B NOWAK Stress Principles CASA Seminar 8th March 2006


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2

3

22

2

22

1

222

2

3

2

2

2

1

nnn

nnn

IIIIIISN

IIIIIIN

1232

2

2

1 nnn


If we consider again the state of stress at P referenced to principal axes and we let the principal stresses be

ordered according to I> II> III. As before, we may express Nand Son any plane at P in terms of the components of

the normal to that plane by the equationsn

which, along with condition

provide us with three equations for the three direction cosines n1, n

2, and n

3. Solving these equations, we obtain:

IIIIIIIII

SIININ

IIIIIIII

SINIIIN

IIIIIII

SIIINIIN

n

n

n

22

3

22

2

22

1

In these equations, I, II, and IIIare known; Nand Sare functions of the direction cosines ni.



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Graphical interpretation



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Plane Stress

0000

0

2221

1211

ij


000

00

00

)2(

)1(

*

ij

212

22112211

)2(

)1(

22



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Deviator and Spherical Stress

M

M

M

ij

00

00

00

B. NOWAK, Stress Principles., CASA Seminar, 8 March 2006

Mean Normal Stress

Spherical State of Stress

Every kind of Stress can be decomposed into a spherical portion and a portion S ijknown as

the deviator stress in accordance with the equation:

iiM 3

1

3

1332211

kkijijMijijij SS 3

1



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Numerical example of stress analysis


Lets consider a thin shell plate given on the figure:

Dimensions:

0.2x0.08x0.001

Materials property:

E=2e11Pa, v=0.3

Type of analysis:Plain Stress

B. NOWAK, Stress Principles., CASA Seminar, 8th March 2006


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We take into consideration two types of fixation and one type

of load:

Totally fixed edge

Degree of freedom in Y

(2) direction is not fixed

Case B

Case A

Uniformly applied

pressure 1MPa



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, p , ,

Case A



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, p , ,

Case A



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Case A



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Case B



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Case B



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Case B



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(implanted femur bone)

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