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, .
These constraint conditions are equivalent to the concentrated force BC.
Assume that . This satisfies the traction
BCs on and equation (34). Therefore,
That means is independent of . Therefore,
in order to satisfy the BCs, , i.e.,
Checking for compatibility, , we get
The general solution is
The only non-zero stress is .
Plugging into equation (33), we get
Plugging into equation (32), we get
The stress state is
Special Case : [edit | edit source]
A concentrated point load acting on a half plane.
Plug in ,
Plug into ,
To fix the rigid body motion, we set when ,
and set when and .Then,
The displacements are singular at and .
Is the small strain assumption satisfied ?