### Saint-Venant’s Principle, Interpretations and Applications by Kotur Raghavan

**General**

Adhemar Jean Claude Barre de
Saint-Venant, better known as Saint-Venant, was a nineteenth century
Mechanician and Mathematician. He has made pioneering contributions in the
areas of fluid Mechanics and Theory of Elasticity.

Fig. 1

In solid mechanics he is best
remembered for development of mathematical equations applicable to torsion and
the Saint-Venant’s Principle which deals with treatment of statically
equivalent forces. This article is essentially concerned with Saint-Venant’s
Principle, interpretations and applications.

**The Principle**

**If the forces acting on a small portion of the
surface of an elastic body are replaced by another statically equivalent
system of forces acting on the same portion of the surface, this
redistribution of loading produces substantial changes in the stresses locally
but has a negligible effect on the stresses at distances which are large
in comparison with the linear dimensions of the surface over which the
forces are acting.**

There are three
phrases of importance here which are underlined. The first is ‘statically
equivalent’. This implies that the new systems of forces have the same resultant
force and moment as the earlier one. The other two phrases together imply that
the stresses at locations ‘sufficiently’ away from the load application region
remain the same in an engineering sense.

The meaning of this
can be understood with the help of two simple case study problems.

**Case Study – 1, Tension strip**

A tension strip is
taken up and three different statically equivalent forces are applied at the
free end. Details are given in the Fig. 2 below.

Fig. 2

The resultant force in each of
the three cases is 40 newtons applied at the midpoint. The contours of axial
sress are given in the figure 3 below.

Fig. 3

For easy comparison all the three
contours are drawn to the same stress range of zero to 2.5. The nominal
far-field stress is unity and this is observed to be the computed value in all
the three cases away from the load application region.

Fig. 4

The variations of axial stress
along the line AB, which is from the free end to the fixed end, are shown superimposed
in Fig. 4 above. An important observation is that the stresses for the three
loading conditions remain the same after a distance of about 40 mm. It is no
coincidence that 40 mm happens to be the width of the tension strip. This is
exactly what the Saint-Venant principle tells us. If the width were to 6o mm
the stresses would have agreed with one another after a distance of 60 mm.

**Case Study 2 – Table Top**

In this case study, an abuse
condition of a large office table is discussed. The problem is that of a man
using the table as a ladder for reaching the ceiling fan. Such a situation is
not uncommon. He may stand with feet as
support. He may stand with toes as support. He may even sit. These situations
are analysed and compared.

Fig. 5

Details of the problem are given
in Fig. 5. Regions P, S1 and S2 denote three loading conditions. Point P
corresponds to extreme loading condition of point load (close to the case of
person standing on toes). S1 is a square region 300*300 which would simulate
the person standing in normal posture on the table at the centre. The region S2
is a square of size 600*600 and this would simulate the condition of the person
sitting on the table. The table top is made of wood having elasticity modulus
of 10000 MPa. The thickness is 18 mm.

Only the table top is modelled using
shell elements. Constraints are applied at the table top-leg interfaces.
Because of symmetry, only one quarter is analysed. A load of 1000 newtons is
considered in the analysis and this would simulate the gravity load of a person
weighing about 110 kilograms.

Fig. 6

Details of the finite element
model are shown in Fig. 6 above. In addition to the constraints indicated, all the
in-plane translations are constrained. Because only a quarter of the full
domain is considered, a load of 250 newtons is applied.

Fig. 7

The contours of von Mises stress
for the load distributed over S1 region are shown in Fig. 7 above. The contours
will be same on either the top surface or the bottom surface.

Fig. 8

The variations of von Mises
stress along the line joining C1 to C2 for the three loading conditions are
shown superimposed in Fig. 8 above. The effect of the mode of load application
on the stress is seen to cease after a distance of about 250 mm. This is
comparable to the representative dimension of the region S2. It is to be noted
that the high stress at C1 for point load application (P) is spurious and is
caused by singularity.

**Saint-Venant Principle – A Corollary**

An offshoot of the principle
applies to self-equilibrated loads. It can be stated as

**A self-equilibrated
systems of tractions result in stresses that decay rapidly with ****distance from the region where tractions are
applied and vanishes at points which are ‘sufficiently’ away.**

We can get a feel for this by
doing this experiment. Place your hand on the table and place a few books on
the back of the palm. The palm gets compressed and you feel the pain locally in
the palm. Now lift the hand along with the books. Pain will now be felt over
rest of the hand all the up to the shoulders. When the hand is on the table the
applied load was locally reacted to (short load path) and there is local
equilibrium. But when the hand is lifted the reaction the reaction point is
elsewhere and the load path is longer.

We will now get insight into this
with the help of a case study involving overlapping plates.

Fig. 9

In Fig. 9 is shown the model
taken up for analysis. This is a two-dimensional representation of two
overlapping plates clamped together at discrete points. The clamp forces are
simulated in the form of surface pressures as shown. The applied loads or in
self-equilibrium. Clamping force transmitted through contacts as shown.

Fig. 10

In Fig. 10, the contours of
compressive stress are shown. The red coloured regions have near-zero stress.

Fig. 11

The variation of the compressive
stress along the interface in one of the plates is shown in Fig. 11 above. The
regions of non-zero stress are longer than the pressure application zone due to
the so-called pressure cone (also called the cone of compression) effect.
However, the stress outside the pressure cone regions is consistent with the
Saint-Venant principle.

**Summary and Recommendations**

·
The Saint-Venant principle has been demonstrated
from two different perspectives. In both the forms, the principle can be exploited
for simplification of finite element models.

·
The analyst has keep in mind two aspects while
making use of the principle to simplify the model. They are the purpose of
analysis and the region of interest or the results of importance. In addition,
of course, some amount of engineering judgment will be necessary.

·
Application of statically equivalent loads
becomes handy to overcome many load simulation challenges. For example, the
load transfer from a shaft to the bearing is quite complex in oil film as well
as in rolling element bearings. Rigorous model will be needed only when the
local stresses or of interest. Otherwise, it will suffice to apply the load in
the form of uniform pressure or even nodal forces.

·
Regarding self-equilibrated loads, the problem
considered in this article is typical of threaded fasteners. In any bolted
joint, the bolt tension due to preload is in equilibrium with compressive
forces in the joint members. For studying the local effects it will be
sufficient to analyse one fastener assembly. The domain should cover the cone
of compression. The results are insensitive to far-field displacement
constraints.

My response to your very important compilation

ReplyDeleteSaint Venant's Principle I think is much more fundamental and embodies an important law of nature that any disturbance would die down exponentially, i.e., its effect would only be a local. I am not sure whether it is made use of any other branch of mechanics..

It reminds me of the definition of Principle of Minimum Potential energy ( which we use in Elasticity )is also a very fundamental law of nature. Euler's statement says "Since the fabric of universe is most perfect and is the work of a most wise creator nothing whatsoever takes place in the universe in which some relation of maximum or minimum does not occur" (History of Strength of Materials -Timoshenko). In other words the nature around us is the laziest (least efforts meaning minima) and yet most efficient (highest output meaning maxima)

Saint Venant's principle as we understand is an adopted version of more fundamental law.

The best example of statically equivalent loading is that of a section subjected to pure moment which is realized by four point loading.. The ripples around load points vanish very quickly and we get the desired state of stress in the rest of the section

Thanks for valuable inputs. I will modify by including the important observations. Saint-Venant had a phenomenally great insight. He was a great mathematician too.

DeleteI have read all the comments and suggestions posted by the visitors for this article are very fine, we will wait for your next article so only. Thanks!

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