ECAR 202 Verification Problems

Overview

ECAR 202 is an INL internal report describing a set of simulations used to verify the quality of Abaqus solutions. The problems described here are taken from ECAR 202 and are used to verify the quality of Bison solutions.

An accuracy tolerance of 3% is used in ECAR 202.

Stress Due to Thermal Strain

The problem is a cantilever beam constrained from axial displacement at both ends.

$var element = document.getElementById("moose-equation-bc22226f-62eb-4727-b069-934461f422b4");katex.render(" s_{xx} = -E \\alpha \\Delta T", element, {displayMode:true,throwOnError:false});$ where

E is Young's modulus (30E6)
$var element = document.getElementById("moose-equation-0b616f22-7991-4c9c-863e-508147dca783");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is the coefficient of thermal expansion (6E-6)
$var element = document.getElementById("moose-equation-4d7c747d-09a9-4990-b41a-ddd3c817c1ec");katex.render("\\Delta T", element, {displayMode:false,throwOnError:false});$ is the change in temperature (500/3)

This gives $var element = document.getElementById("moose-equation-67a2e71e-fb0d-4756-beb5-3cfd4e47afef");katex.render(" s_{xx} = 30000", element, {displayMode:true,throwOnError:false});$

Bison computes 30000.

Thick-walled Cylinder

This problem is a thick-walled cylinder with an internal radius of 2, an external radius of 3, and a height of 4. The number of radial elements is 8, and the number of axial elements is 12.

The analytic solution for hoop stress at the inner radius, $var element = document.getElementById("moose-equation-aa1c3d92-f9f3-4fcb-a076-be850d5a14dd");katex.render("s_{zz}", element, {displayMode:false,throwOnError:false});$ , is (Harvey, 1974) $var element = document.getElementById("moose-equation-2072f8d5-b3dd-449f-9799-e6198a2967a5");katex.render(" s_{zz} = P_i\\frac{r_o^2+r_i^2}{r_o^2-r_i^2}", element, {displayMode:true,throwOnError:false});$ where

$var element = document.getElementById("moose-equation-e3b6d3e3-9790-44fe-bb3f-fcf7d7e59e30");katex.render("P_i", element, {displayMode:false,throwOnError:false});$ is the internal pressure (20000)
$var element = document.getElementById("moose-equation-1b9aea09-4947-4214-ae9b-6feca9f42e0a");katex.render("r_i", element, {displayMode:false,throwOnError:false});$ is the inner radius
$var element = document.getElementById("moose-equation-99b9e96b-3114-482a-bebf-1d1128d9e970");katex.render("r_o", element, {displayMode:false,throwOnError:false});$ is the outer radius

This gives $var element = document.getElementById("moose-equation-a29cca1d-e056-477d-aa65-8bc2636545f5");katex.render(" s_{zz} = 52000", element, {displayMode:true,throwOnError:false});$

With linear quadrilateral elements, Bison computes this hoop stress as 52084, which is an error of 0.16%.

With 8-node quadratic quadrilateral elements, Bison computes this hoop stress as 52089, which is an error of 0.17%.

References

John F. Harvey. Theory and Design of Modern Pressure Vessels. Van Nostrand Reinhold, New York, second edition, 1974.

@book{HARVEY1974,
    author = "Harvey, John F.",
    title = "Theory and Design of Modern Pressure Vessels",
    publisher = "Van Nostrand Reinhold",
    edition = "Second",
    year = "1974",
    address = "New York"
}

Overview
References