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Elasticity/Concentrated force on half plane

Concentrated force on a half-plane

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


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


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


Superpose the two solutions. The stresses are

The displacements are


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