ENGINEERING
STRENGTH OF
MATERIALS II
(MST21A)
1. Normal stress distribution
2. Sai nt Venant ’s pr i nci pl e
3. Stress concentration
What you should know
Before you start with this lecture, you should be able to
do the following:
1. Draw a free body diagram
2. Apply the conditions for static equilibrium
Expected outcomes
Upon completion of this lecture, you should be able to do
the following:
1. To represent the stress due to applied load on a section
2. To understand and use Saint Venant’s principle for
calculation of stress applied on a section
3. To identify regions of stress concentration
Background
1. Stress is a means of measuring the force distribution within
a body
2. Strain is a means of measuring a body’s deformation
3. The mathematical relationship between stress and strain
depends on the type of material from which the body is
made.
4. If the material behaves in a linear elastic manner, then
Hooke’s law applies, and there is a proportional relationship
between stress and strain.
1. Normal stress distribution in axially loaded bars
If a load P is applied at
one free end of a prismatic
bar,
1. The once horizontal and
vertical grid lines drawn on
the bar become distorted,
and localized deformation
occurs at each end
2. Throughout the midsection
of the bar, the lines remain
horizontal and vertical.
Normal stress distribution in axially loaded bars
• For axially loaded bars,
we
assume
average
UNIFORM STRESS over
the cross-sectional area.
• Considering the manner
in which a rectangular
bar will deform elastically
when
the
bar
is
subjected to the force P
applied
along
centroidal axis:
its
Stresses in axially loaded bars
3. A variation of stress distribution occur at sections a–a, b–b,
and c–c, as shown.
4. If the material remains elastic, then the strains caused by
this deformation are directly related to the stress in the bar
through Hooke’s law, 𝜎 = 𝐸𝜀.
Stresses in axially loaded bars
3. By comparison, the stress tends to reach a uniform value at
section c–c, which is sufficiently removed (located far) from
the end since the localized deformation caused by P
vanishes.
4. The minimum distance from the bar’s end where this occurs
can be determined using a mathematical analysis based on
the theory of elasticity.
5. It has been found that this distance should at least be equal
to the largest dimension of the loaded cross section.
6. Hence, section c–c should be located at a distance at least
equal to the width (not the thickness) of the bar.
2. Saint-Venant’s principle
7. The fact that the localized stress and deformation behave in
this manner is known as the Saint-Venant’s principle.
8. The Saint-Venant’s principle states that:
The stress and strain produced at a point in a body, at a
sufficient certain distance from the point of application of the
external load P, will be the same as the stress and strain
produced any applied external load that has the same
statically equivalent RESULTANT, and being applied to the
body within the same region.
In other words, Saint-Venant’s principle states that if the
forces acting on a small portion of an elastic body are
replaced by a statically equivalent system of forces on the
same portion of the surface, the effect upon the stresses in
the body is negligible except in the immediate area affected
by the applied forces. The stress field remains unchanged in
areas of the body which are relatively distant from the
surfaces upon which the forces are changed. Statically
equivalent systems implies that the two distributions of
forces have the same resultant force and moment.
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Saint-Venant’s principle
For example, if two symmetrically applied forces P / 2 act on
the bar, the stress distribution at section c–c will be the same
and therefore equivalent to 𝜎𝑎𝑣𝑔 =
𝑃
𝐴
3.
Stress concentration in axially loaded bars
1. Discontinuities in a bar such as
holes, grooves etc. result in a change
of stress at these regions
2. Stresses may be higher in these
regions
3. These regions of higher stress is
known as stress concentrations
4. Stress concentrations can also occur
at point loads
application.
at the point of
Stress concentration in axially loaded bars
Zone of stress
concentration
Simulation with The Finite
Element Method (FEM)
allows to visualize the stress
distribution along the bar.
The red color indicates zones
(regions) with higher stress
(Stress concentration) usually
in transition zones such as
corners, fillets or holes.