Elasticity/Concentrated force on half plane

Concentrated force on a half-plane

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Concentrated force on a half plane

From the Flamant Solution

 

and

 

If   and , we obtain the special case of a concentrated force acting on a half-plane. Then,

 

or,

 

Therefore,

 

The stresses are

 

The stress   is obviously the superposition of the stresses due to   and  , applied separately to the half-plane.


Problem 1 : Stresses and displacements due to F2

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The tensile force   produces the stress field

 
 
Stress due to concentrated force   on a half plane

The stress function is

 

Hence, the displacements from Michell's solution are

 

At  , ( ,  ),

 

At  , ( ,  ),

 

where

 

Since we expect the solution to be symmetric about  , we superpose a rigid body displacement

 

The displacements are

 

where

 

and   on  .

Problem 2 : Stresses and displacements due to F1

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The tensile force   produces the stress field

 
 
Stress due to concentrated force   on a half plane

The displacements are

 

Stresses and displacements due to F1 + F2

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Superpose the two solutions. The stresses are

 

The displacements are