VM-WB-MECH-062

VM-WB-MECH-062
Stresses in a Long Cylinder

Overview

Reference:	Timoshenko, S. (1956). Strength of Material, Part II: Elementary Theory and Problems (3rd ed., pp. 213, Problem 1; p. 213, Article 42). New York, NY: D. Van Nostrand Co., Inc.
Solver(s):	Ansys Mechanical
Analysis Type(s):	Static Structural Analysis
Element Type(s):	Axisymmetric Shell

Test Case

A long thick-walled cylinder is initially subjected to an internal pressure p. Determine the radial displacement δ_r at the inner surface, the radial stress σ_r, and tangential stress σ_t, at the inner and outer surfaces and at the middle wall thickness. Internal pressure is then removed and the cylinder is subjected to a rotation ω about its center line. Determine the radial σ_r and tangential σ_t stresses at the inner wall and at an interior point located at r = X_i.

Figure 83: Schematic

Material Properties

Geometric Properties

E = 30 x 10⁶ psi

ν = 0.3

ρ = 0.281826 lbm/in³

a = 4 in

b = 8 in

X_i = 5.43 in

Case 1:

Pressure = 30,000 psi (radial direction)

Case 2:

Rotational velocity = 1000 rad/s (Y direction)

Analysis Assumptions and Modeling Notes for Ansys Mechanical

Because the geometry and loading are symmetric about the y-axis, the above problem can be analyzed as an axisymmetric problem in Ansys Mechanical.

Figure 84: Schematic for Ansys Mechanical

Case 1:

Case 2:

Results Comparison for Ansys Mechanical

Results		Target	Mechanical	Error (%)
Case 1:	Displacement_r, in (r = 4 in)	0.0078666	0.0076267	-3.050
	Stress_r, psi (r = 4 in)	-30000	-29988	-0.040
	Stress_r, psi (r = 6 in)	-7778.0	-7775.3	-0.035
	Stress_t, psi (r = 8 in)	0	0.79611	--
	Stress_t, psi (r = 4 in)	50000	49988	-0.024
	Stress_t, psi (r = 6 in)	27778	27775	-0.011
	Stress_t, psi (r = 8 in)	20000	19999	-0.005
Case 2:	Stress_t, psi (r = 4 in)	40588	41672	2.671
	Stress_r, psi (r = 5.43 in)	4753.0	4933.7	3.802
	Stress_t, psi (r = 5.43 in)	29436	29719	0.961