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

B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Stress
and
Stress Principles
CASA Seminar
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Basic Overview
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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
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Stress definitions
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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
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Stress
(Basic assumptions and definitions)
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In continuum mechanics a body is considered
stress free if the only forces present are those interatomic 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).
fi
bi
pi
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
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:
 ave 
 ave 
m
V
m
V
and the density ρ at point P by the limit of this ratio
as the volume shrinks to the point:
m dm

V  0 V
dV
  lim
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:
bi  pi
b  p
The density is in general a scalar function of
position and time:
  (x, t )
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Cauchy Stress Principle
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.
fi
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:
f i
df i
( nˆ )


t
i
*
0 S
dS *
lim
*
S
M i
0
*
 0 S
lim
*
S
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
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:
t (eˆ1 )  t1(eˆ1 )eˆ1  t 2(eˆ1 )eˆ 2  t3(eˆ1 )eˆ 3
t (eˆ 2 )  t1(eˆ 2 )eˆ1  t 2(eˆ 2 )eˆ 2  t3(eˆ 2 )eˆ 3
ˆ
ˆ
ˆ
ˆ
t (e3 )  t1(e3 )eˆ1  t 2(e3 )eˆ 2  t3(e3 )eˆ 3
or using summation convention:
ˆ
ˆ
t (ei )  t (jei )eˆ j
(i  1,2,3)
This equation expresses the stress vector at P point for a given coordinate plan in terms of its rectangular Cartesian
components.
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
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
nˆ  nieˆ i
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,
* ti(nˆ ) dS  *ti(eˆ1 ) dS1  *ti(eˆ 2 ) dS2  *ti(eˆ 3 ) dS3  *bi dV  0
Now, taking into consideration area surfaces, volume we can rewrite above:
(eˆ )
* ti(nˆ ) dS  *ti j dS  *bi dV  0
1
( eˆ )
* ti(nˆ )  *ti j n j  * bi h  0
3
Now, letting the tetrahedron shrink to point P we get :
( eˆ )
ti(nˆ )  ti j n j
or by defining
( eˆ j )
 ji  ti
ti( nˆ )   ji n j
t ( nˆ )  nˆ  σ
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
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:
t
( nˆ )
1
, t 2( nˆ ) , t3( nˆ )

 11  12  13 
 n1 , n2 , n3  21  22  23 
 31  32  33 
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.
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).
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Principal Stresses, Principal Stress Directions
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:
ti( nˆ )   ji n j
t ( nˆ )  nˆ  σ
and the substitution property of the Kronecker delta allows to rewrite:

ji
  ij n j  0
In 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,
ni ni  1
For non-trivial solutions the determinant of coefficients on nj must vanish. That is,
 ji   ij  0
which upon expansion yields a cubic in σ (called the characteristic equation of the stress tensor)
 3  I  2  II   III  0
whose roots σ(1), σ(2), σ(3) are the principal stress values of σij. The coefficients are
the first, second, and third invariants, respectively, of σij and may be expressed in terms of its components by:
known
as
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Principal Stresses, Principal Stress Direction
I   ii  trσ

 
1
 ii jj   ij ji   1 trσ 2  tr σ 2
2
2
III   ijk 1i 2 j 3k  det σ
II 
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.
ti(nˆ )  ni
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 t (nˆ ) of to the principal planes, there are no shear stresses acting in these planes.
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Normal and Shear Stress Components
The stress vector on an arbitrary plane at P may be resolved into a component normal to the plane having a magnitude σN,
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,  N  ti(nˆ ) ni and in as much as ti(nˆ )   ij n j , it
follows that:
 N   ij n j n i
or
 N  t (nˆ )nˆ
Also, from the geometry of decomposition, we get:
 S  ti( nˆ )ti( nˆ )   N2
 S  t (nˆ )  t (nˆ )   N2
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Stress Matrix Components
Let’s consider:
Shear stress components
 ij*  aiq a jm qm
or
σ*  Aσ A T
4
3
 3
0

57
0
24


5

 5 0
5
 ij*   0 1 0   0 50 0   0 1
4
 4
3 

 24 0 43 
0
0
5 
 5
 5
 
4
25 0 0 
5  
0    0 50 0 
3
  0 0 75
5 
Normal stress components
Principal stress components
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Mohr’s Circles for Stress
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 nˆ to that plane by the equations
 N   I n12   II n22   III n32
 N2   S2   I2 n12   II2 n22   III2 n32
which, along with condition
n12  n22  n32  1
provide us with three equations for the three direction cosines n 1, n2, and n3. Solving these equations, we obtain:
 N   II  N   III  S2
 I   II  I   III 
   III  N   I  S2
n22  N
 II   III  II   I 
   I  N   II  S2
n32  N
 III   I  III   II 
n12 
In these equations, σI , σII, and σIII are known; σN and σS are functions of the direction cosines ni.
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Mohr’s Circles for Stress
Graphical interpretation
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Plane Stress
0
 (1)
  0  ( 2 )
 0
0
0
0
0
 11  12
  21  22
 0
0
0
0
0
 
*
ij
 
ij
 (1)   11   22
 11   22
2




12
 ( 2) 
2
2
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Deviator and Spherical Stress
Mean Normal Stress
M 
1
 11   22   33   1  ii
3
3
Spherical State of Stress
 
ij
 M
  0
 0
0
M
0
0 
0 
 M 
Every kind of Stress can be decomposed into a spherical portion and a portion Sij known as
the deviator stress in accordance with the equation:
1
3
 ij  Sij   ij M  S ij   ij kk
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
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
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
We take into consideration two types of fixation and one type
of load:
Case A
Degree of freedom in Y
(2) direction is not fixed
Uniformly applied
pressure 1MPa
Case B
Totally fixed edge
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
Case A
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
Case A
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
Case A
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
Case B
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
Case B
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
Case B
B. NOWAK, Stress – Principles…., CASA Seminar, 8th March 2006
Numerical example of stress analysis
(implanted femur bone)