Curved beam elasticity theory based on the displacement function method using a finite difference scheme

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 Boundary conditions of the computational model

Boundary	Boundary conditions
Boundary	Normal component	Tangential component
Right Boundary \(\theta = \theta _{i}\)	\(u_{r}(r,\theta _{i}) = 0\)	\(u_{\theta } (r,\theta _{i}) = 0\)
Left Boundary \(\theta = \theta _{\max } = \theta _{i} + \theta _{e}\)	\(u_{r}(r,\theta _{\max } ) = 0\)	\(u_{\theta } (r,\theta _{\max } ) = 0\)
Inner Boundary \(r = r_{\mathrm{ib}}\)	\(\sigma _{r}(r_{\mathrm{ib}},\theta ) = 0\)	\(\tau _{r\theta } (r_{\mathrm{ib}},\theta ) = 0\)
Out Boundary \(r = r_{\mathrm{ob}}\), θ ≤ 90^∘	\(\sigma _{r}(r_{\mathrm{ob}},\theta ) = - q(\lambda \cos \theta + \sin \theta )\)	\(\tau _{r\theta } (r_{\mathrm{ob}},\theta ) = - q(\cos \theta - \lambda \sin \theta )\)
Out Boundary \(r = r_{\mathrm{ob}}\), θ>90^∘	\(\sigma _{r}(r_{\mathrm{ob}},\theta ) = - q( - \lambda \cos \theta + \sin \theta )\)	\(\tau _{r\theta } (r_{\mathrm{ob}},\theta ) = - q(\cos \theta + \lambda \sin \theta )\)