R0031(2): Wrapped thick cylinder under pressure and thermal loading

This problem provides evidence that Abaqus can reproduce the result from the benchmark defined by NAFEMS and cited as the reference solution.

This page discusses:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/Explicit

Elements tested

C3D8R

C3D20

CSS8

S4R

S8R

SC8R

Problem description

Mesh:

One-quarter of the cylinder cross-section and half of the length is modeled. The same problem is analyzed with different meshes. Meshes using linear solids and shells consist of eight elements along the radial and axial directions. Meshes using quadratic solids and general-purpose shells consist of four elements along the radial and axial directions. The models using continuum elements are stacked with two single-layer elements through the thickness in Abaqus/Standard and with four single-layer elements in Abaqus/Explicit. The models using the S4R and S8R elements use a composite section definition. The continuum shell models use two modeling techniques to model the cylinder in the thickness direction: (1) a single element with a composite section and (2) four single-layer elements stacked through the thickness.

Material:

Inner isotropic cylinder: E = 2.1E5 MPa, $ν$ = 0.3, $α$ = 2.0 × 10⁻⁵/°C.

Outer circumferentially wound cylinder: $E_{1}$ = 130 GPa, $E_{2}$ = 5 GPa, $E_{3}$ = 5 GPa, $ν_{12}$ = 0.25, $ν_{13}$ = 0.25, $ν_{23}$ = 0, $G_{12}$ = 10 GPa, $G_{13}$ = 10 GPa, $G_{23}$ = 5 GPa, $α_{11}$ = 3.0 × 10⁻⁶/°C, $α_{22}$ = 2.0 × 10⁻⁵/°C, $α_{33}$ = 2.0 × 10⁻⁵/°C.

Boundary conditions:

$u_{z}$ = 0 at z = 0.

Loading:

Case 1: Internal pressure of 200 MPa.

Case 2 (Abaqus/Standard only): Internal pressure of 200 MPa + temperature rise of 130°C.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test R0031/2 from NAFEMS publication R0031, “Composites Benchmarks,” Issue 2, February 5, 2001.

Results and discussion

The results are given in Table 1 and Table 2. All results at $z = 0$ . The values enclosed in parentheses are percentage differences with respect to the reference solution.

Table 1. Abaqus/Standard analysis.
Analysis	Case 1				Case 2
	inner cylinder		outer cylinder		inner cylinder		outer cylinder
	$σ_{θ}$ at r = 23	$σ_{θ}$ at r = 25	$σ_{θ}$ at r = 25	$σ_{θ}$ at r = 27	$σ_{θ}$ at r = 23	$σ_{θ}$ at r = 25	$σ_{θ}$ at r = 25	$σ_{θ}$ at r = 27
Composite S8R	1421 (−9.2%)	1421 (–0.6%)	878 (0.4%)	878 (15.7%)	1241 (−10.1%)	1241 (−1.5%)	1056 (–3.7%)	1056 (12.8%)
Stacked C3D20	1567 (0.1%)	1432 (0.2%)	880 (0.6%)	756 (−0.4%)	1382 (0.1%)	1262 (0.2%)	1062 (–3.1%)	933 (−0.3%)
Composite SC8R	1477 (−5.6%)	1477 (3.3%)	900 (2.9%)	900 (18.6%)	1299 (−5.9%)	1299 (3.1%)	1078 (-1.6%)	1078 (15.2%)
Stacked SC8R	1552 (–0.9 %)	1470 (2.8%)	849 (−2.9%)	787 (3.7%)	1450 (5.0%)	1319 (4.7%)	1011 (−7.8%)	935 (−0.1%)
Composite SC8R*	1477 (–5.6%)	1477 (3.3%)	900 (2.9%)	900 (18.6%)	1299 (–5.9%)	1299 (3.1%)	1078 (–1.6%)	1078 (15.2%)
Stacked SC8R*	1587 (1.4%)	1491 (4.3%)	848 (–3.1%)	785 (3.4%)	1452 (5.1%)	1318 (4.6%)	1011 (–7.8%)	935 (–0.1%)
Stacked CSS8	1600 (2.2%)	1406 (–1.7%)	876 (0.1%)	732 (–3.6%)	1419 (2.8%)	1245 (–1.2%)	1060 (–3.3%)	904 (–3.4%)
Reference solution	1565	1430	875	759	1381	1260	1096	936

* Abaqus/Standard results with enhanced hourglass control.

Table 2. Abaqus/Explicit analysis.
Analysis	Case 1
	inner cylinder		outer cylinder
	$σ_{θ}$ at r = 23	$σ_{θ}$ at r = 25	$σ_{θ}$ at r = 25	$σ_{θ}$ at r = 27
Composite S4R	1417 (−9.5%)	1427 (–0.2%)	879 (0.5%)	879 (15.8%)
Stacked C3D8R	1548 (−1.1%)	1479 (3.5%)	850 (−2.8%)	788 (3.8%)
Reference solution	1565	1430	875	759

Input files

Abaqus/Standard input files

nco2s8rx.inp: Composite S8R model.
nco2c3d2.inp: Stacked C3D20 model.
r312_std_sc8r_composite.inp: Composite SC8R model.
r312_std_sc8r_stacked.inp: Stacked SC8R model.
r312_std_sc8r_composite_eh.inp: Composite SC8R model with enhanced hourglass control.
r312_std_sc8r_stacked_eh.inp: Stacked SC8R model with enhanced hourglass control.
nco2c3d2_css8_exp.inp: Stacked CSS8 model.

Abaqus/Explicit input files

r312shl.inp: Composite S4R model.
r312sol.inp: Stacked C3D8R model.