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SAINT VENANT’S PRINCIPLE

By

VIJAYA LAKSHMI EMANI.

MEEN – 5330.

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INTRODUCTION[3]

Saint-Venant’s 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 Venant’s principle used to justify this approach.

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DEFINITIONS[2]

SAINT VENANT’S 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 VENANT’S PRINCIPLE – MATHEMATICAL FORM[1]

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' < UR", where UR' and UR" are strain energies.

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COMMON ENGINEERING INTERPRETATION OF THE

SAINT VENANT’S PRINCIPLE[5]

“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-Venant’s 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 PRINCIPLE[4]

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 = 02. Torsion if P = 0 and n x M = 03. Bending by a torque if P = 0 and n . M = 04. 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 CALCULATION[4]

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

1. . T = 0 }in the volume V of the beam.2. x S x = 0

3. n . T =0 on the lateral surface, where n.k=0

4. k . T x k = 05. ∫s dS k . T = 0 on the surfaces z=±L r being equal to xi+yj.6. ∫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

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

yxkk

E 2

2

2

21 )(

2

2

2

2

jiijjjxy

yx

2

T33

332

2

Tx 0332

2

Ty

033

2

Tyx

0332

2

2

2

T

yx

dSxs dS

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DEFORMATION CALCULATION

∫s dS r x (k . T) = ± Mj we obtain on z=L,

-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

MjByAxydSiBxyAxdS ss )()( 22

11 kkxEI

MS

rxzkxEI

MS 1

2

11

22 rzixrxzk

EI

221

22 irxrkxz

iz

EI

Ms

kxzxyjyxzi

EI

M 222

2

1

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PRACTICAL APPLICATIONS AND LIMITATIONS

APPLICATIONSSaint-Venant’s 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.

LIMITATIONSThe Saint Venant’s 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 Venant’s 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 z=L, and to a force –Rk acting at a point p' symmetrical to P with respect to plane z=0. solve this problem using Saint Venant’s 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/elastprins/elastprins.htm7.http://www.engin.brown.edu/courses/En224/svtorsion/svtorsion.html