4.2.2 LE2: Cylindrical shell bending patch test

Products: ABAQUS/Standard ABAQUS/Explicit

Elements tested

S3 S3R S3RS S4 S4R S4R5 S4RS S4RSW S8R S8R5 S9R5

STRI3 STRI65 SC6R SC8R

Problem description

Model:

Sector of cylindrical shell with a thickness = 0.01 m.

Material:

Linear elastic, Young's modulus = 210 GPa, Poisson's ratio = 0.3, density = 7800 kg/m³.

Boundary conditions:

Edge AB is clamped. Axial displacements are constrained along edges AD and BC.

Loading:

Uniform normal edge moment of 1000/unit length along edge DC. In the explicit dynamic analysis the loading is applied such that a quasi-static solution is obtained.

Reference solution

This is a test recommended by the National Agency for Finite Element Methods and Standards (U.K.): Test LE2 from NAFEMS publication TNSB, Rev. 3, “The Standard NAFEMS Benchmarks,” October 1990.

Stress: Outer surface tangential stress at point E is 60 MPa.

Results and discussion

The results shown in Table 4.2.2–1 through Table 4.2.2–4 are interpolated from the integration points to the required nodal location. The values enclosed in parentheses are percentage differences with respect to the reference solution.

Table 4.2.2–1 ABAQUS/Standard analysis, 30°.

Element	Bottom Surface (MPa)	Top Surface (MPa)
S3/S3R	–46.113 (–23%)	43.342 (–28%)
S4	–62.617 (4%)	52.925 (–12%)
S4R	–57.319 (–4%)	57.479 (–4%)
S4R*	–54.848 (–9%)	54.302 (–9%)
S4R5	–57.934 (–3%)	58.118 (–3%)
S8R	–50.698 (–16%)	50.330 (–16%)
S8R5	–57.764 (–4%)	58.103 (–3%)
S9R5	–57.828 (–4%)	58.232 (–3%)
STRI3	–36.007 (–40%)	36.093 (–40%)
STRI65	–53.883 (–10%)	53.770 (–10%)
SC6R	–43.5 (–27.5%)	40.5 (–32.5%)
SC8R	–58.5 (–2.5%)	58.7 (–2.2%)
SC8R*	–55.936 (–6.7%)	–55.25 (–7.9%)
*ABAQUS/Standard results with enhanced hourglass control.

Table 4.2.2–2 ABAQUS/Explicit analysis, 30°.

Element	Bottom Surface (MPa)	Top Surface (MPa)
S3R	–44.9 MPa (–25%)	42.9 MPa (–28%)
S3RS	–43.4 MPa (–28%)	43.9 MPa (–27%)
S4R	–55.6 MPa (–7.3%)	55.6 MPa (–7.3%)
S4RS	–58.7 MPa (–2.1%)	58.9 MPa (–1.8%)
S4RSW	–57.8 MPa (–3.7%)	57.9 MPa (–3.5%)

These results vary significantly from the target value since the mesh is too coarse to capture a curvature of

30°. The mesh can be refined easily by reducing the arc angle to

10°. The following results show that such mesh refinement greatly improves the accuracy of the results.

Table 4.2.2–3 ABAQUS/Standard analysis, 10°.

Element	Bottom Surface (MPa)	Top Surface (MPa)
S3/S3R	–59.701 (–0.50%)	59.406 (–0.99%)
S4	–60.416 (0.69%)	59.389 (–1.02%)
S4R	–59.929 (–0.12%)	59.927 (–0.12%)
S4R*	–59.926 (–0.12%)	59.912 (–0.12%)
S4R5	–59.996 (–0.01%)	59.995 (–0.01%)
S8R	–59.650 (–0.58%)	59.704 (–0.49%)
S8R5	–59.895 (–0.18%)	60.005 (0.01%)
S9R5	–59.666 (–0.56%)	60.064 (0.12%)
STRI3	–60.534 (0.89%)	60.482 (0.80%)
STRI65	–59.600 (–0.67%)	59.671 (–0.55%)
SC6R	–59.957 (–0.07%)	59.635 (–0.6%)
SC8R	–60.053 (0.09%)	60.053 (0.09%)
SC8R*	–60.05 (0.08%)	–60.04 (0.07%)
*ABAQUS/Standard results with enhanced hourglass control.

Table 4.2.2–4 ABAQUS/Explicit analysis, 10°.

Element	Bottom Surface (MPa)	Top Surface (MPa)
S3R	–59.6 MPa (–0.67%)	59.3 MPa (–1.17%)
S3RS	–59.5 MPa (–0.83%)	59.1 MPa (–1.50%)
S4R	–59.9 MPa (–0.17%)	59.9 MPa (–0.17%)
S4RS	–60.0 MPa (–0.01%)	60.0 MPa (–0.01%)
S4RSW	–59.9 MPa (–0.17%)	59.9 MPa (–0.17%)