Given:
A wedge of infinite length with a concentrated load
per
unit wedge thickness at the vertex. Plane stress/strain.
Wedge loaded transversely by a concentrated load
|
Find:
The stress field in the wedge.
From the Flamant solution, we know that the stress field in the wedge is
![{\displaystyle {\begin{aligned}\sigma _{rr}&={\frac {2}{r}}\left(C_{1}\cos \theta +C_{2}\sin \theta \right)\\\sigma _{r\theta }&=0\\\sigma _{\theta \theta }&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0606a8906ed8711e4ca8ab135f88c7cc2c6d8874)
The constants
and
can be found by using the equilibrium
conditions
![{\displaystyle {\begin{aligned}2\int _{-\beta }^{\beta }\left(C_{1}\cos \theta -C_{2}\sin \theta \right)\cos \theta ~d\theta &=0\\P+2\int _{-\beta }^{\beta }\left(C_{1}\cos \theta -C_{2}\sin \theta \right)\sin \theta ~d\theta &=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2100c1d6b5c46d043383d72b3522dbfae674525b)
or,
![{\displaystyle {\begin{aligned}C_{1}\left[2\beta +\sin(2\beta )\right]&=0\\P+C_{2}\left[\sin(2\beta )-2\beta \right]&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a39e53f232f6f2f66c9044a485cc264b3c5ce4b)
Therefore,
![{\displaystyle C_{1}=0~;~~C_{2}={\frac {P}{2\beta -\sin(2\beta )}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee9d2dbbf5fa157fead8bf7c97e5d768104511a)
Hence, the stresses are
![{\displaystyle {\begin{aligned}\sigma _{rr}&={\frac {2P\sin \theta }{r\left[2\beta -\sin(2\beta )\right]}}\\\sigma _{r\theta }&=0\\\sigma _{\theta \theta }&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2733d8e3fdd09c065548909a28b97b1732a41d2)