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| Basics of Pipe Stress Analysis: The Principal Stresses |
The Principal Stresses
When we
calculate the stresses, we choose a set of orthogonal directions and define the
stresses in this coordinate system. In a pipe subjected to internal pressure or
any other loads, the most used choice of the coordinate system is one
comprising of
- Axial Or Longitudinal direction stress, SL
- Circumferential Or Hoop Stress, SH
- Radial Directional Stress, SR
These Stress which stretch or compress a
grain/crystal are called normal stress, because of act normally to the surface
of the crystal. But all grains are not oriented as the grain is shown in the
diagram. In fact, the grain would have been oriented in the pipe wall in all
possible orientations. The above stress would also have stress components in
the direction normal to the face of such randomly oriented crystals. Each
crystal thus does face normal stress. One of these orientations must be such
that it maximizes one of the normal stresses. It also called Major Principal
Stress. The mechanics of solid-state that, it would also be an orientation,
which minimizes some other normal stress. It also called Minor Principal
Stress.
Normal stress for such orientation (Maximum/Major
Normal stress and Minimum/Minor Normal Stress) are called principle stress and
are designated as S1 (Maximum), S2 & S3 (Minimum).
The Sum
of the three Normal Stress for all orientation is always the same for any given
external load.
SL+SH+SR=S1+S2+S3
Importance of principal stresses
Assume
that a material can be deemed to fail of any normal stress exceed some
threshold value. If a Conventional coordinate system is used, one may find for
certain stress that SL, SH & SR are within this threshold limit. The design
then would appear to be safe. However, grins that are oriented in maximum normal
stress orientation may have one of the stresses more than this threshold. The
pipe would thus fail as far as these grains are concerned. The design has to be
safe for such a worst-case. Principal stresses are thus a way of defining
worst-case scenarios as far as normal stress is concerned.
Shear Stresses
In addition
the normal stresses, grain can be subjected to shear stresses which act
parallel to the crystal surface as against the perpendicular direction
applicable for normal stresses. Shear stresses occur if the pipe is subjected
to Torsion, Bending, etc. Just as there is an orientation on which normal
stresses are maximum, there is an orientation that maximizes shear stresses.
The maximum shear stress in the 3-D state of stress can be shown as
I.e. half of the difference between the maximum (S1) &
minimum (S3) principal stress.
The maximum shear stress is important to calculate because
failure may occur due to shear stress. Hence, it is necessary to take the
worst-case scenario for shear Stress also as above and ensure against failure.
Hence,
- It is easy to defined stress in the coordinate system such as Axial-Hoop-Radial (L-H-R) that we defined for the pipe. The load-bearing cross-section is then well defined. The stress components are calculated as the ratio of the load to the bearing cross-section.
- Similarly, it is possible to calculate shear stress in a particular plan for the given torsion or bending loads
- The principal stresses and strain can be calculated from the normal stress and shear stress in an orthogonal coordinate system.
Complex Stresses in Piping System
In most pipe design cases, the radial
component of normal stress is negligible as compared to the other two components.
The 3-D state of this can be simplified to the 2-D state of the stress. Use of
Mohr’s Circle then allows calculating the Principal Stresses & Maximum
shear stresses as follows.
The third principal stress (S3) is zero.
(minimum/negligible)
All failure theories state that these principals or maximum shear stress or some
combination of them should be within allowable limits for the MOC under
consideration.
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