Cylindrical Shell with Free Edges at a Temperature Gradient across the Thickness (in the Radial Direction)

Objective: Determination of the stress-strain state of a cylindrical shell with free edges subjected to a temperature gradient across the thickness.

Initial data file: 4_33.spr

Problem formulation: The cylindrical thin-walled shell free from constraints is subjected to a temperature gradient across the thickness. The temperatures of the cylinder wall on its internal t₁ and external surfaces t₂ are constant. The temperature varies linearly across the thickness of the wall. Determine the stress tensor components on the internal and external surfaces of the shell in the meridian σ_x^ext (σxint) and circumferential σ_φ^ext (σ_φ^int)  directions, as well as the radial displacements w.

References: S.P. Timoshenko, Theory of Plates and Shells. — Moscow: OGIZ. Gostekhizdat, 1948.

Initial data:

Finite element model: Design model – general type system, shell elements – 12800 four-node elements of type 44. The spacing of the finite element mesh in the meridian direction is 0.025 m  and in the circumferential direction is 4.5º. The dimensional stability of the design model is provided by imposing constraints according to its symmetry conditions. Number of nodes in the design model – 12880.

Results in SCAD

Design model

Deformed model

Values of radial displacements w (mm)

Values of radial displacements w (mm) for the fragment of the model from the section with the central angle of 18.0º

Values of stresses on the external surface of the shell in the meridian direction σ_x^ext (kN/m²)

Values of stresses on the internal surface of the shell in the meridian direction σ_x^int (kN/m²)

Values of stresses on the external surface of the shell in the circumferential direction σ_φ^ext (kN/m²)

Values of stresses on the internal surface of the shell in the circumferential direction σ_φ^int (kN/m²)

Comparison of solutions:

Notes: In the analytical solution the stresses on the internal and external surfaces of the shell in the meridian σ_x^ext (σ_x^int) and circumferential σ_φ^ext (σ_φ^int) directions, as well as the radial displacements w can be determined according to the following formulas (S.P. Timoshenko, Theory of Plates and Shells. — Moscow: OGIZ. Gostekhizdat, 1948, p. 399), which give a good approximation “at points at a considerable distance from the edges of the shell”:

\[ w=0.5\cdot \alpha \cdot \left( {t_{1} -t_{2} } \right)\cdot a\cdot \sqrt {\frac{1+\nu }{3\cdot \left( {1-\nu } \right)}} \cdot e^{-\beta \cdot x}\cdot \left( {\sin \left( {\beta \cdot x} \right)-\cos \left( {\beta \cdot x} \right)} \right); \] \[ \sigma_{x}^{ext} =\frac{E\cdot \alpha \cdot \left( {t_{1} -t_{2} } \right)}{2\cdot \left( {1-\nu } \right)}\cdot \left[ {-1+e^{-\beta \cdot x}\cdot \left( {\cos \left( {\beta \cdot x} \right)+\sin \left( {\beta \cdot x} \right)} \right)} \right]; \] \[ \sigma_{x}^{int} =\frac{E\cdot \alpha \cdot \left( {t_{1} -t_{2} } \right)}{2\cdot \left( {1-\nu } \right)}\cdot \left[ {1-e^{-\beta \cdot x}\cdot \left( {\cos \left( {\beta \cdot x} \right)+\sin \left( {\beta \cdot x} \right)} \right)} \right]; \] \[ \sigma_{\phi }^{ext} =\frac{E\cdot \alpha \cdot \left( {t_{1} -t_{2} } \right)}{2\cdot \left( {1-\nu } \right)}\cdot \left[ {-1+\nu \cdot e^{-\beta \cdot x}\cdot \left( {\cos \left( {\beta \cdot x} \right)+\sin \left( {\beta \cdot x} \right)} \right)-\sqrt {\frac{1-\nu^{2}}{3}} \cdot e^{-\beta \cdot x}\cdot \left( {\sin \left( {\beta \cdot x} \right)-\cos \left( {\beta \cdot x} \right)} \right)} \right]; \] \[\sigma_{\phi }^{int} =\frac{E\cdot \alpha \cdot \left( {t_{1} -t_{2} } \right)}{2\cdot \left( {1-\nu } \right)}\cdot \left[ {1-\nu \cdot e^{-\beta \cdot x}\cdot \left( {\cos \left( {\beta \cdot x} \right)+\sin \left( {\beta \cdot x} \right)} \right)-\sqrt {\frac{1-\nu^{2}}{3}} \cdot e^{-\beta \cdot x}\cdot \left( {\sin \left( {\beta \cdot x} \right)-\cos \left( {\beta \cdot x} \right)} \right)} \right], where: \]\[ \beta =\sqrt[4]{\frac{3\cdot \left( {1-\nu^{2}} \right)}{a^{2}\cdot h^{2}}}. \]