In this article Stress Linearization Procedure based on Annex 5-A of ASME Sec VIII Div 2 code will be illustrated using an example.
Example Problem: A carbon steel vessel (ID 500 mm x 20 thk x 1000 mm long) has a nozzle connection (ID 250 mm x 20 thk x 130 mm projection) of the same material. The ends of the vessel are fixed and a shear load of 1000 kN is applied at the nozzle to shell junction along the vessel axis. Using FEA the linearized Von-Mises Membrane and Membrane + Bending Stresses in the vessel needs to be evaluated near the location of maximum stress.
The problem analysis is conducted using ANSYS.
ANSYS Analysis Plots
The Finite Element Model
Boundary Conditions and Loads (Note: Both ends including the end not visible in the plot are fixed)
Von-Mises Stress in the vessel
Coordinate system to define the path for the Stress Classification Line (SCL). A local coordinate system is created (using create coordinate system aligned with hit point normal method in ANSYS) near the max Von-Mises Stress location, so that its X-Y-Z axes represent tangential (radial), hoop and normal (meridional) directions respectively.
SCL Path (Near the Location of Max Von-Mises Stress)
Stress Linearization Procedure:
The hoop, normal (meridional) and tangential (radial), direct along with corresponding shear stress components were recorded across 5 equidistant points along the SCL path.
Subscript Meanings: h – hoop, n – normal, t – tangential, hn – hoop normal, nt – normal tangential, th – tangential hoop
Thickness across SCL Path, t = 20 mm
Distance between points, Δx = t/4 = 5 mm
Calculation of Membrane Stress Tensor
The membrane stress tensor is the tensor comprised of the average of each stress component along the stress classification line.
Membrane Hoop Stress
Subscripts: hm – hoop membrane, hnm – hoop normal membrane etc
Calculation of Bending Stress Tensor
(a) Bending stresses are calculated only for the local hoop and meridional (normal) component stresses, and not for the local component stress parallel to the SCL or in-plane shear stress.
(b) The linear portion of shear stress needs to be considered only for shear stress distributions that result in torsion of the SCL (out-of-plane shear stress in the normal-hoop plane).
(c) The bending stress tensor is comprised of the linear varying portion of each stress component along the stress classification line
Bending Hoop Stress
Subscripts: hb – hoop bending, hnb – hoop normal bending etc
Note that bending for other components are not calculated (see points (a) and (b)) as required by ASME Sec VIII Div 2, Annex 5-A para 5-A.4.1.2. The default linearization routine of ANSYS or other FEA softwares will calculate bending for all components hence do not comply with Annex 5-A requirement. However some softwares eg. Abaqus allow users to select which stress components to include in the linearization routine, thus giving the user the flexibility to comply with the Annex 5-A requirements.
Calculation of Membrane + Bending Stress Tensor
x = 0 represents 1st point on the SCL
x = t represents the other end point on the SCL
Subscripts: hmb0 – hoop membrane + bending at x = 0, hnmbt – hoop normal membrane + bending at x = t etc
Calculation of Linearized Von-Mises Membrane and Membrane + Bending Stress
Subscripts: eqvm – Equivalent Membrane Stress or Von-Mises Membrane Stress, eqvmb0 – Equivalent Membrane + Bending Stress at x=0, eqvmbt – Equivalent Membrane + Bending Stress at x=t
Reference: Composite Simpson’s Rule