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.

This page discusses:

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.