LE2: Cylindrical shell bending patch test

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

The following topics are discussed:

Elements tested
Problem description
Reference solution
Results and discussion
Input files

ProductsAbaqus/StandardAbaqus/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 t= 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 1 through Table 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 1. Abaqus/Standard analysis, $θ =$ 30°.
Element	Bottom Surface (MPa)	Top Surface (MPa)
S3/S3R	−44.3 (–26%)	40.6 (–32%)
S4	−63.2 (5%)	54.0 (–10%)
S4R	−58.0 (–3%)	58.0 (–3%)
S4R*	–55.0 (–8%)	55.2 (–8%)
S4R5	−58.6 (–2%)	58.6 (−2%)
S8R	−50.7 (−16%)	50.4 (−16%)
S8R5	−57.8 (−4%)	58.2 (−3%)
S9R5	−57.9 (−4%)	58.3 (−3%)
STRI3	−37.9 (−37%)	36.0 (−40%)
STRI65	−53.6 (−11%)	53.9 (−10%)
SC6R	–43.9 (−27%)	43.9 (−27%)
SC8R	−59.7 (−1%)	59.7 (−1%)
SC8R*	–54.8 (–9%)	–54.8 (–9%)
*Abaqus/Standard results with enhanced hourglass control.

Table 2. Abaqus/Explicit analysis, $θ =$ 30°.
Element	Bottom Surface (MPa)	Top Surface (MPa)
S3R	−43.2 MPa (−28%)	39.7 MPa (−34%)
S3RS	−44.7 MPa (−26%)	42.2 MPa (−30%)
S4R	−58.2 MPa (−3%)	58.3 MPa (−2.8%)
S4RS	−57.0 MPa (−5%)	56.9 MPa (−5.2%)
S4RSW	−57.3 MPa (−4.5%)	57.4 MPa (−4.3%)

These results vary significantly from the target value since the mesh is too coarse to capture a curvature of $θ =$ 30°. The mesh can be refined easily by reducing the arc angle to $θ =$ 10°. The following results show that such mesh refinement greatly improves the accuracy of the results.

Table 3. Abaqus/Standard analysis, $θ =$ 10°.
Element	Bottom Surface (MPa)	Top Surface (MPa)
S3/S3R	−60.1 (0.2%)	59.9 (−0.2%)
S4	−60.5 (0.8%)	59.5 (−0.8%)
S4R	−60.0 (0%)	60.0 (0%)
S4R*	–60.0 (0%)	60.0 (0%)
S4R5	−60.0 (0%)	60.0 (0%)
S8R	−59.6 (−0.7%)	59.7 (−0.5%)
S8R5	−59.9 (−0.2%)	60.0 (0%)
S9R5	−59.7 (–0.5%)	60.0 (0%)
STRI3	−60.8 (1.3%)	60.8 (1.3%)
STRI65	−59.6 (−0.7%)	59.7 (−0.5%)
SC6R	–60.2 (0.3%)	60.2 (0.3%)
SC8R	–60.2 (0.3%)	60.2 (0.3%)
SC8R*	–60.2 (0.3%)	60.2 (0.3%)
*Abaqus/Standard results with enhanced hourglass control.

Table 4. Abaqus/Explicit analysis, $θ =$ 10°.
Element	Bottom Surface (MPa)	Top Surface (MPa)
S3R	−60.2 MPa (−0.3%)	59.9 MPa (−0.1%)
S3RS	−60.0 MPa (0%)	59.8 MPa (−0.3%)
S4R	−60.0 MPa (0%)	60.0 MPa (0%)
S4RS	−60.1 MPa (−0.1%)	60.1 MPa (−0.1%)
S4RSW	−60.0 MPa (0%)	59.9 MPa (−0.2%)

Input files

Abaqus/Standard input files

θ = 30°:

nle2xf3c.inp: S3/S3R elements.
nle2xe4c.inp: S4 elements.
nle2xf4c.inp: S4R elements.
nle2xf4c_eh.inp: S4R elements with enhanced hourglass control.
nle2x54c.inp: S4R5 elements.
nle2x68c.inp: S8R elements.
nle2x58c.inp: S8R5 elements.
nle2x59c.inp: S9R5 elements.
nle2x63c.inp: STRI3 elements.
nle2x56c.inp: STRI65 elements.
nle2_std_sc6r_30.inp: SC6R elements.
nle2_std_sc8r_30.inp: SC8R elements.
nle2_std_sc8r_30_eh.inp: SC8R elements with enhanced hourglass control.

θ = 10°:

nle2xf3f.inp: S3/S3R elements.
nle2xe4f.inp: S4 elements.
nle2xf4f.inp: S4R elements.
nle2xf4f_eh.inp: S4R elements with enhanced hourglass control.
nle2x54f.inp: S4R5 elements.
nle2x68f.inp: S8R elements.
nle2x58f.inp: S8R5 elements.
nle2x59f.inp: S9R5 elements.
nle2x63f.inp: STRI3 elements.
nle2x56f.inp: STRI65 elements.
nle2_std_sc6r_10.inp: SC6R elements.
nle2_std_sc8r_10.inp: SC8R elements.
nle2_std_sc8r_10_eh.inp: SC8R elements with enhanced hourglass control.

Abaqus/Explicit input files

θ = 30°:

le2_s3r_c.inp: S3R elements.
le2_s3rs_c.inp: S3RS elements.
le2_s4r_c.inp: S4R elements.
le2_s4rs_c.inp: S4RS elements.
le2_s4rsw_c.inp: S4RSW elements.

θ = 10°:

le2_s3r_f.inp: S3R elements.
le2_s3rs_f.inp: S3RS elements.
le2_s4r_f.inp: S4R elements.
le2_s4rs_f.inp: S4RS elements.
le2_s4rsw_f.inp: S4RSW elements.