Elasticity/Axially loaded wedge

Axially Loaded Wedge

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Elastic wedge loaded by an axial force

The BCs at   are

 

What about the concentrated force BC?

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  • What is   at the vertex ?
  • The traction is infinite since the force is applied on zero area. Consider equilibrium of a portion of the wedge.

At  , the BCs are

 

For equilibrium,  . Therefore,

 

These constraint conditions are equivalent to the concentrated force BC.

Solution Procedure

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Assume that  . This satisfies the traction BCs on   and equation (34). Therefore,

 

Hence,

 

That means   is independent of  . Therefore, in order to satisfy the BCs,  , i.e.,

 

Checking for compatibility,  , we get

 

The general solution is

 

Therefore,

 

The only non-zero stress is  .

 

Plugging into equation (33), we get

 

Hence,

 

Plugging into equation (32), we get

 

Therefore,

 

The stress state is

 

Special Case

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A concentrated point load acting on a half plane.

 

Displacements

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where

 

Plug in  ,

 

Plug   into  ,

 

Hence,

 

Solving,

 

Therefore,

 

To fix the rigid body motion, we set   when  , and set   when   and  .Then,

 

The displacements are singular at   and  . At  ,

 

Is the small strain assumption satisfied ?