SAINT VENANTS PRINCIPLE

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SAINT VENANTS PRINCIPLE

• By
• VIJAYA LAKSHMI EMANI.
• MEEN 5330.

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INTRODUCTION3

• Saint-Venants principle is used to justify
  approximate solutions to boundary value problems
  in linear elasticity.
• For example, when solving problems
  involving bending or axial deformation of slender
  beams and rods, one does not prescribe loads in
  any detail. Instead, the resultant forces acting
  on the ends of a rod is specified, or the
  magnitudes of point forces acting on a beam.
  Saint Venants principle used to justify this
  approach.

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DEFINITIONS2

• SAINT VENANTS PRINCIPLE
• If a system of forces acting on a small portion
  of the surface of an elastic body is replaced by
  another statically equivalent system of forces
  acting on the same portion of the surface, the
  redistribution of loading produces substantial
  changes in the stresses only in the immediate
  neighborhood of the loading, and the stresses are
  essentially the same in the part of the body
  which are at large distances in comparision with
  the linear dimension of the surface on which the
  forces are changed.
• STATICALLY EQUIVALENT
• It means that the two distributions of forces
  have the same resultant force and moment.

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SAINT VENANTS PRINCIPLE MATHEMATICAL FORM1

• Let S' and S'' be two non intersecting sections
  both outside a sphere B. If the section S'' lies
  at a greater distance than the section S' from
  the sphere B in which a system of self
  equilibrating forces P acts on the body, then
• UR' lt UR", where UR' and
  UR" are strain energies.

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COMMON ENGINEERING INTERPRETATION OF THE SAINT
VENANTS PRINCIPLE5 As long as the different
approximations are statically equivalent , the
resulting solutions will be valid provided we
focus on regions sufficiently far away from the
support.That is, the solutions may significantly
differ only within the immediate vicinity of the
support. OR It is to say that the manner in
which the forces are distributed over a region is
important only in the vicinity of the region.
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  EXAMPLE.6 Surface loaded half-space.
Closed form expressions are known for the
fields induced by many axi-symmetric traction
distributions acting on the surface of a
half-space. For example, the fields down the
symmetry axis due to a uniform normal pressure
acting on a circular region of radius a are
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EXAMPLE CONTINUED Where P is the resultant force
acting on the loaded region. Similarly, the
stresses due to a Hertz pressure
Are, Expand these in a/z and one will
find that to leading order in a/z both
expressions are identical. Indeed, the leading
order term is the stress induced by a point force
acting at the origin. Far from the loaded region,
the two traction distributions induce the same
stresses, because the resultant force acting on
the loaded region is identical.
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GENERAL SAINT VENANT BEAM PROBLEM

• It is very complicated to solve the beam
  problem by considering some definite distribution
  of surface forces on the end sections of the
  beam. Hence we assume the validity of
  Saint-Venants principle.
• So now the exact distribution of the surface
  forces on each end section is replaced by another
  one that is statically equivalent-that is, one
  with the same resultant R and the same moment M
  with respect to some point O.

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GENERAL CASES TO SOLVE USING SAINT VENANT
PRINCIPLE4

• There are four general classes of
  Saint-Venant problem corresponding to various
  choices of end loading.
• 1. Extension if M 0
  and n x P 0
• 2. Torsion if P
  0 and n x M 0
• 3. Bending by a torque if P 0 and n
  . M 0
• 4. Bending by a force if M 0 and n
  . P 0

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BENDING OF A BEAM BY A TORQUE

• Consider a cylindrical beam of arbitrary
  cross section and a length 2L.On the end sections
  are applied normal forces, distributed in such a
  way that on each end section the resultant is
  zero, and the total moment is tangent to the
  section.

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SYSTEM OF EQUATIONS TO BE CONSIDERED FOR
DEFORMATION CALCULATION4

• We assume that the torque applied at the end
  section z L is Mj. Then for equilibrium we must
  have a torque Mj on end section z -L. The
  system of Equilibrium equations to be satisfied
  are then
• ? . T 0 in the volume V of the
  beam.
• ? x S x ? 0
• 3. n . T 0 on the lateral
  surface, where n.k0
• k . T x k 0
• ?s dS k . T 0 on the surfaces zL r
  being equal to xiyj.
• ?s dS r x (k . T) Mj

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DEFORMATION CALCULATION

• From the equations n . T 0 and k . T x k
  0
• The stress dyadic has the simple form as
  follows
• T T33 kk.
• Also from ? . T 0
• we get ?/?z( T33) 0.This shows that T33
  depends on x and y only.
• Thus the Strain dyadic is given by
• S 1 kk - ?(ii jj) T33(x,y).
• E

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DEFORMATION CALCULATION
T33

• S ii
  
• The compatibility equation ? x S x ? gives the
  following four equations
• 0
  
• Which implies that T33 Ax By C
• where A,B and C are arbitrary constants to
  be determined from the boundary conditions. From
  condition 5.?s dS k . T 0 we get
• ?s dS ( Ax By C) 0.
• Also from ?s ds r 0 we have 0 and
  y 0 .
• From ?s dS r x (k . T) Mj

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DEFORMATION CALCULATION

• ?s dS r x (k . T) Mj we obtain on zL,
• -j
  
• The exact solution for any cross section is
  T-M/I(xkk)
• Where I ?sdS x2
• The strain dyadic is
  
• Or
• -
  
• From this we get the displacement vector is
• 
  

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PRACTICAL APPLICATIONS AND LIMITATIONS

• APPLICATIONS
• Saint-Venants principle is used to justify
  approximate solutions to boundary value problems
  in linear elasticity.  
• The principle of Saint Venant allows us to
  simplify the solution of many problems by
  altering the boundary conditions while keeping
  the systems of applied forces statically
  equivalent. A satisfactory approximate solution
  can be obtained.

• LIMITATIONS
• The Saint Venants Principle, as enunciated in
  terms of the strain energy functional, does not
  yield any detailed information about individual
  stress components at any specific point in an
  elastic body. But such information is clearly
  desired .
• Saint-Venant himself limited his principle to the
  problem of extension, torsion and flexure of
  prismatic and cylindrical bodies.

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HOMEWORK

• 1. Write a one page essay on Saint Venants
  Principle.
• 2. A cylindrical beam of length 2L has an
  arbitrary cross section. The beam is subjected to
  a force Rk acting at a point P of the boundary of
  the end section zL, and to a force Rk acting at
  a point p' symmetrical to P with respect to plane
  z0. solve this problem using Saint Venants
  principle and determine the deformation.

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REFERENCES

• 1.Foundations of Solids Mechanics by Y.C.Fung.
• 2.Elasticity Theory and Applications Second
  Edition by Adel S. Saada.
• 3.Theory of Elasticity by Timoshenko and
  Goodier.
• 4.Introduction to Elasticity by Gerard Nadeau.
• 5.Theory of Elasticity by Southwell.
• 6.http//www.engin.brown.edu/courses/En222/Notes/e
  lastprins/elastprins.htm
• 7.http//www.engin.brown.edu/courses/En224/svtorsi
  on/svtorsion.html