VE0066 | Plastic Thick-Walled Vessel

Description

A thick-walled vessel is loaded by inner pressure, which is chosen so that the vessel reaches the elastic-plastic state. The problem is modeled as a quarter model. While neglecting self-weight, determine and compare the analytical and numerical solution for the radial position of the plastic zone border r_y under the Tresca hypothesis for the yield surface.

Material	Elastic-Plastic	Modulus of Elasticity	E	200000.000	MPa
		Poisson´s Ratio	ν	0.250	-
		Yield Strength	f_y	200.000	MPa
Geometry		Inner Radius	r₁	200.000	mm
Geometry		Outer Radius	r₂	300.000	mm
Load		Inner Pressure	p₁	80.000	kPa

Plastic Thick-Walled Vessel

Analytical Solution

The analytical solution of the given problem is analogous to the analytical solution of VE0064 - Thick-Walled Vessel and VE0065 - Two-Layered Thick-Walled Vessel.

The stress state of the thick-walled vessel is described by the equation of equilibrium

\frac{d σ_{r}}{d r} + \frac{σ_{r} - σ_{t}}{r} = 0

σ_r	Radial stress
σ_t	Tangential stress

Equilibrium Equation for Thick-Walled Vessel

The Tresca criterion implies the tensile yield strength f_y to be equal to

f_{y} = σ_{t} - σ_{r}

Tresca Criterion for Thick-Walled Vessel

which then with the boundary condition σ_r=-p₁ renders the equation of equilibrium into the relation

σ_{r} (r) = f_{y} \ln \frac{r}{r_{1}} - p_{1}

Radial Stress

The relationship between the pressure p_y at the yield radius r_y follows:

p_{y} = p_{1} - f_{y} \ln \frac{r_{y}}{r_{1}}

Pressure at Yield Radius

Further, the elastic part of the vessel has to be described. Again from the Tresca criterion another formula for the pressure at the yield radius results: