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

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

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

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

• 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).

fi
bi
pi
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Stress (Density definition)
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006

• 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

and the density ? at point P by the limit of
this ratio as the volume shrinks to the point
The density is in general a scalar function of
position and time
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
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Cauchy Stress Principle
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
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 ?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 ?Sequivalent to
a resultant force ?fi and a resultant moment ?Mi
at 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
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The Stress Tensor(rectangular Cartesian
components)
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
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
or using summation convention
This equation expresses the stress vector at P
point for a given coordinate plan in terms of its
rectangular Cartesian components.
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The Stress Tensor(analysis for arbitrary
oriented plane)
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
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,
Now, taking into consideration area surfaces,
volume we can rewrite above
Now, letting the tetrahedron shrink to point P we
get
or by defining
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The Stress Tensor
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
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 sji 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 sji 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
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 s11, s22, and s33 are called normal
stresses. The six arrows lying in the coordinate
planes and pointing in the directions of the
coordinate axes, namely, s12, s21, s23, s32, s31,
and s13 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).
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Principal Stresses, Principal Stress Directions
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
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
In the three linear homogeneous equations
expressed implicitly above, the tensor components
sij are assumed known the unknowns are the three
components of the principal normal ni, and the
corresponding principal stress s. 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,
which upon expansion yields a cubic in s (called
the characteristic equation of the stress tensor)
whose roots s(1), s(2), s(3) are the principal
stress values of sij. The coefficients are known
as the first, second, and third invariants,
respectively, of sij and may be expressed in
terms of its components by
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Principal Stresses, Principal Stress Direction
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Because the stress tensor sij 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 s is called a
principal stress value of sij . 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.
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Normal and Shear Stress Components
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
The stress vector on an arbitrary plane at P may
be resolved into a component normal to the plane
having a magnitude sN, along with a shear
component which acts in the plane and has a
magnitude sS, as shown in figure. (Here, sN and
sS 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 sN is
given by the dot product, and in
as much as , it follows that
Also, from the geometry of decomposition, we get
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Stress Matrix Components
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Lets consider
Shear stress components
Normal stress components
Principal stress components
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Mohrs Circles for Stress
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
If we consider again the state of stress at P
referenced to principal axes and we let the
principal stresses be ordered according to sI gt
sII gt sIII. As before, we may express sN and sS
on any plane at P in terms of the components of
the normal to that plane by the equations
which, along with condition
provide us with three equations for the three
direction cosines n1, n2, and n3. Solving these
equations, we obtain
In these equations, sI , sII, and sIII are known
sN and sS are functions of the direction cosines
ni.
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Mohrs Circles for Stress
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Graphical interpretation
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Plane Stress
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006

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Deviator and Spherical Stress
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Mean Normal Stress
Spherical State of Stress
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
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Lets consider a thin shell plate given on the
figure Dimensions 0.2x0.08x0.001 Materials
property E2e11Pa, v0.3 Type of
analysis Plain Stress
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
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
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Case A
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Case A
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Case A
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Case B
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Case B
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Numerical example of stress analysis
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006
Case B
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Numerical example of stress analysis (implanted
femur bone)
B. NOWAK, Stress Principles., CASA Seminar,
8th March 2006