Introduction to Elasticity/Axially loaded wedge

From Wikiversity
Jump to navigation Jump to search

Axially Loaded Wedge[edit | edit source]

Elastic wedge loaded by an axial force

The BCs at are

What about the concentrated force BC?[edit | edit source]

  • 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[edit | edit source]

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 : [edit | edit source]

A concentrated point load acting on a half plane.

Displacements[edit | edit source]

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 ?