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

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

Stress

and

Stress Principles

CASA SeminarCASA Seminar

Basic Overview

• Stress Definitions• Cauchy Stress Principle• The Stress Tensor• Principal Stresses, Principal Stress Direction• Normal and Shear Stress Components• Mohr’s Circles for Stress• Special Kinds of Stress• Numerical Examples of Stress Analysis


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


Stress (Basic assumptions and definitions)

• In continuum mechanics a body is considered stress free if the only forces present are those inter-atomic forces required to hold the body together

• Basic types of forces are distinguished from one another:

– Body forces i.e.: gravity, inertia; designated by vector symbol bi (force per unit mass) or pi (force per unit volume); acting on all volume elements, and distributed throughout the body;

– Surface forces i.e.: pressure; denoted by vector symbol

fi (force per unit area of surface across they which they act); act upon and are distributed in some fashion over a surface element of the body, regardless of whether that element is part of the bounding surface, or an arbitrary element of surface within the body;

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

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


bi

pi

fi

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 ΔV whose mass is ΔM. We define the average density of this volume element by the ratio:

V

mave


and the density ρ at point P by the limit of this ratio as the volume shrinks to the point:

dV

dm

V

mV

0lim

V

mave

The units of density are kilograms per cubic meter (kg/m3). Two measures of body forces, bi having units of Newtons per kilogram (N/kg), and pi having 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 of position and time:

),( tx

Cauchy Stress Principle

)ˆ(

**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 fi and 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 ΔS* 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 ΔS*equivalent to a resultant force Δfi and a resultant moment ΔMi at P.

The Cauchy stress principle asserts that in the limit as the area ΔS* shrinks to zero with P remaining an interior point, we obtain:

0lim*0*

S

M i

S

fi

The Stress Tensor(rectangular Cartesian components)


We can introduce a rectangular Cartesian reference frame at P, there is associated with each of the area elements 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)ˆ(

32)ˆ(

21)ˆ(

1)ˆ(

3)ˆ(

32)ˆ(

21)ˆ(

1)ˆ(

3)ˆ(

32)ˆ(

21)ˆ(

1)ˆ(

ˆˆˆ

ˆˆˆ

ˆˆˆ

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 ˆ)ˆ()ˆ( )3,2,1( i

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 of a 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 ˆˆ

0***** 3)ˆ(

2)ˆ(

1)ˆ()ˆ( 321 dVbdStdStdStdSt iiiii eeen

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

0*** )ˆ()ˆ( dVbdStdSt iiij en

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

jii ntt j )ˆ()ˆ( en

0*3

1** )ˆ()ˆ( hbntt ijii

j en

or by defining

)ˆ( j

iji t e

σnt n

n

ˆ)ˆ(

)ˆ(jjii nt

The Stress Tensor


The Cauchy stress formula expresses the stress vector associated with the element of area having an outward normal ni at point P in terms of the stress tensor components σji at 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, or equivalently, the nine stress tensor components σji at 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)ˆ(

3)ˆ(

2)ˆ(

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 σ11, σ22, and σ33 are called normal stresses. The six arrows lying in the coordinate planes and pointing in the directions of the coordinate axes, namely, σ12, σ21, σ23, σ32, σ31, and σ13 are called shear stresses.

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:

σnt n

n

ˆ)ˆ(

)ˆ(jjii nt

0 jijji nIn the three linear homogeneous equations expressed implicitly above, the tensor components σij are assumed known; the unknowns are the three components of the principal normal ni, and the corresponding principal stress σ. To complete the system of equations for these four unknowns, we use the normalizing condition on the direction cosines,

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

0 ijji

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

023 IIIIIIwhose roots σ(1), σ(2), σ(3) are the principal stress values of σ ij. The coefficients are known as the first, second, and third invariants, respectively, of σij and may be expressed in terms of its components by:

Principal Stresses, Principal Stress Direction


σ

σσ

σ

det2

1

2

1

321

22

kjiijk

jiijjjii

ii

III

trtrII

trI

Because the stress tensor σij is asymmetric tensor having real components, the three stress invariants are real, and likewise, the principal stresses being roots are also real.

Directions designated by ni for which above equation is valid are called principal stress directions, and the scalar σ is called a principal stress value of σij . Also, the plane at P perpendicular to ni is referred to as a principal 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 )ˆ(n

)ˆ(nt

Normal and Shear Stress Components

iiN nt )ˆ(n


Also, from the geometry of decomposition, we get:

jiji nt )ˆ(n

nt )n( ˆˆ

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 σ N, along with a shear component which acts in the plane and has a magnitude σS, as shown in figure. (Here, σN and σS are not vectors, 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 σN is given by the dot product, and in as much as , it follows that:

2ˆˆ

2)ˆ()ˆ(

NS

NiiS tt

)n()n(

nn

tt

Stress Matrix Components

T* AAσσ

or

aa qmjmiqij *


Let’s 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

Mohr’s Circles for Stress

23

222

221

222

23

22

21

nnn

nnn

IIIIIISN

IIIIIIN

123

22

21 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 σI > σII > σIII. As before, we may express σN and σS on any plane at P in terms of the components of the normal to that plane by the equationsn̂

which, along with condition

provide us with three equations for the three direction cosines n1, n2, and n3. Solving these equations, we obtain:

IIIIIIIII

SIININ

IIIIIIII

SINIIIN

IIIIIII

SIIINIIN

n

n

n

223

222

221

In these equations, σI , σII, and σIII are known; σN and σS are functions of the direction cosines n i.

Mohr’s Circles for Stress


Graphical interpretation

Plane Stress

000

0

0

2221

1211

ij


000

00

00

)2(

)1(*

ij

212

22112211

)2(

)1(

22

Deviator and Spherical Stress

M

M

M

ij

00

00

00


Mean Normal Stress

Spherical State of Stress

Every kind of Stress can be decomposed into a spherical portion and a portion S ij known as the deviator stress in accordance with the equation:

iiM 3

1

3

1332211

kkijijMijijij SS 3

1

Numerical example of stress analysis


Lets consider a thin shell plate given on the figure:

Dimensions:0.2x0.08x0.001

Material’s property:E=2e11Pa, v=0.3

Type of analysis:Plain Stress



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



Case A



Case B

Numerical example of stress analysis (implanted femur bone)


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

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