SLS 06: Thin cylinder under hydrostatic pressure

Description

Material		Modulus of elasticity	$E =210 000N/mm^2$
Material		Poisson’s ratio	$\nu= 0.3$
Geometry	Cross-section	Thickness	$e=20 mm$
	Boundary conditions	At bottom edge	$R_y$ fixed
		In point A	$R_x$ fixed
		In point B	$R_z$ fixed
Loads		On all plates	Triangular load varying from $0kN/m^2$ at the bottom to $20.0kN/m^2$ at the top.
Mesh		Maximum element size	$0.05m$
Mesh		Minimum element size	$0.00m$

Handcalculation

$\sigma_1(y)=\sigma_{xx}= \frac{q_0 \cdot R \cdot y}{L \cdot e}$

$\sigma_{xx}(2m)=\frac{20kN/m^2 \cdot 1m \cdot 2m}{4m \cdot 0,02m}=0.5N/mm^2$

$\Delta R(y)=\delta_x(y)=\delta_z(y)=\frac{q_0 \cdot R^2 \cdot y}{E \cdot L \cdot e}$

$\delta_x(2m)=\frac{20kN/m^2 \cdot (1m)^2 \cdot 2m}{210000N/mm^2 \cdot 4m \cdot 0,02m}=0.002381mm$

$\Delta y(y)=\delta_y(y)=\frac{q_0 \cdot R \cdot \nu \cdot y^2}{2 \cdot E \cdot L \cdot e}$

$\delta_y(4m)=\frac{20kN/m^2 \cdot 1m\cdot 0.3 \cdot (4m)^2}{2 \cdot 210000N/mm^2 \cdot 4m \cdot 0,02m}=0.002857mm$

Deformation $\delta_y$ in Diamonds

Point	Which result	Independent reference	Diamonds	Difference
Y=2m	Stress $\sigma_xx$	0.5 $N/mm^2$	0.514 $N/mm^2$ ⁽¹⁾	2,80%
Y=4m	Longitudinal deformation $\delta_y$	0,002857mm	0,002847mm	-0,36%
Y=2m	Radial deformation $\delta_x$ and $\delta_z$	0,002381mm	0,002363mm	-0,75%

⁽¹⁾ The stress in Diamonds is determined using a cutline over which the average stress is calculated.

Mécaniciens, S. F. D. (1990). Guide de validation des progiciels de calcul des structures: SSLS 08: cylindre mince sous pression hydrostatique
Roark, R. J., & Young, W. C. (2002). Roark’s Formulas for Stress and Strain (7th edition, Table 13.1 case 1b). McGraw-Hill Companies.

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