SLS 07: Thin cylinder under self-weight

Description

Material		Modulus of elasticity	$E =210 000N/mm^2$
		Poisson’s ratio	$\nu= 0.3$
		Density	$\rho=7850 kg/m^3$ $g=10.0m/s^2$
Geometry	Cross-section	Thickness	$e=2 mm$
	Boundary conditions	At bottom edge	$R_y$ fixed
		In point A	$R_x$ fixed
		In point B	$R_z$ fixed
Loads		Self-weight¹	$\gamma=78.5kN/m^3$
Mesh		Maximum element size	$0.05m$
Mesh		Minimum element size	$0.00m$

¹ The gravitational acceleration $g$ equals $9.81m/s^2$ in Diamonds. You cannot change it. In order to calculate with $g=10.0m/s^2$ , the self-weight of steel was adjusted in Diamonds.

Independent reference results

Open handcalculations

$\sigma_1(z)=\sigma_{zz}=\gamma \cdot z \rightarrow \sigma_1(2m)=78.5kN/m^3 \cdot 2m=0.157N/mm^2$

$\Delta R(z)=\frac{\gamma \cdot \nu \cdot R \cdot z}{E} \rightarrow \Delta R(4m)=\delta_x(0m)=\delta_z(0m)=\frac{78.5kN/m^3 \cdot 0.3 \cdot 1m \cdot 4m}{210000N/mm^2}=0.000449mm$

$\Delta y(z)=\frac{\gamma \cdot z^2}{2 \cdot E} \rightarrow \Delta y(4m)=\delta_y(4m)=\frac{78.5kN/m^3 \cdot (4m)^2}{2 \cdot 210000N/mm^2}=0.002990mm$

Diamonds results and comparison

Deformation $\delta_y$ in Diamonds

Point	Which result	Independent reference	Diamonds	Difference
Y=2m	stress $\sigma_zz$	0,157N/mm²	0,157N/mm²	0,00%
Y=0	Longitudinal deformation $\delta_y$	0,002990mm	0,002991mm	0,02%
Y=0	Radial deformation $\delta_x$ and $\delta_z$	0,000449mm	0,000442mm	-1,46%

References

Mécaniciens, S. F. D. (1990). Guide de validation des progiciels de calcul des structures: SSLS 09: cylindre mince sous son poids propre
Roark, R. J., & Young, W. C. (2002). Roark’s Formulas for Stress and Strain (7th edition, Table 13.1 case 1e). McGraw-Hill Companies.

Tested in Diamonds 2024r01.

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