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Plane ElasticityBy

Tariq Jamil

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ElasticityAn elastic body is defined as one which regains its original

dimensions after the forces acting on it are removed.

Elasticity of a substance depends on the

material possessing linear stress and

strain relations. The range of stress and

strain for which the behavior is linearlyelastic will be known as elastic range.

When the stress exceeds the elastic

limit the object is permanently

distorted and it does not return to its

original shape after the stress isremoved.

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Plane Elasticity Theory

There are 15 independent Equations Stress Equilibrium Equations (3)

Strain-displacement Equations (6)

Stress-Strain Equations (6)

to find out 15 unknown quantities at any point provided following quantities are

adequately defined

Geometry of the body

The boundary Conditions

The body-force field as a function of position

The Elastic Constant

Thus Analytical Solution for the three dimensional elasticity problems are quite difficult

to obtain.

In theory of Elasticity there exist a special class of problems know as Plane Problem

which can be solved more readily than 3D problems due to certain assumptions.

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Plane Elastic Problem

To be Classified as Plane elastic problem the problem must havecertain characteristics as far as Geometry & loading is concerned

GeometryA plane body consists of region of uniform

thickness bounded by two parallel planes

According to Geometry If the thickness t is small as compared to

the dimensions in parallel planes, the

problem is classified as plane Stress

Problem

If the thickness is large compared to the

dimensions in the parallel planes, the

problem is classified as plane strain

problems

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Continue

Loading Body forces, if exist, cannot vary through the thickness of

the region: this is Fx = Fx (x,y) & Fy = Fy(x,y).

Furthermore, the body force in the Z direction must equalto zero.

The surface tractions or loads on the lateral boundary

must be in the plane of the model and must be uniformlydistributed across the thickness i.e. constant in the Zdirection.Hence Tx = Tx (x,y), Ty = Ty(x,y) and Tz = 0

No loads can be applied on the parallel planes bounding

the top and bottom surfaces that is Tn = 0 on Z = t

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Plane Stress Conditions

Stressesz=0 xz=0 yz=0x, y and xymay havenon zero values

Strainsxz=0 yz=0x, y, zand xy may have

non zero values

x

y

xyx

y

z

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Plane Strain Conditions

Stressesxz=0 yz=0x, y, z and xy may havenon zero values

Strainsz= 0 xz=0 yz=0x, yand xymay have

non zero values

x

y

xy

x

y

z

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Governing EquationsPlane Strain

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Continue..

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Plane Stress

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