2.3.1 The barrel vault roof problem

Products: ABAQUS/Standard  ABAQUS/Explicit  

Over the past several years a small set of linear test cases has emerged as a critical test set for shell elements (see, for example, the collection of papers on numerical modeling of shells—edited by Ashwell and Gallagher, 1976—and the survey paper by Belytschko, 1986). The set contains three cases: the barrel vault roof (this example), the cylinder with end diaphragm support subjected to pinching loads (The pinched cylinder problem, Section 2.3.2), and the point loaded hemispherical shell (LE3: Hemispherical shell with point loads, Section 4.2.3). It has been generally accepted that any elements that perform well on all three cases should provide accurate results for most general shell problems. These test cases are included in this manual so that the performance of the shell elements offered in ABAQUS can be assessed.

Most modern shell elements, including those in ABAQUS, do a good job on these problems. Although this is an indication that the elements usually provide good results, it should not be taken as a sufficient demonstration of the quality of an element's performance in all cases. For example, all three of these problems are completely regular geometries; the candidate element's usefulness in irregular geometries (and most practical cases involve a high degree of geometric irregularity) is not tested. In this example we make some attempt to address this issue by modeling not only with the regular mesh that would be the natural choice for the problem, but also with a mesh that might be the basis of analysis of a problem with the same underlying shape but with some type of local, irregular feature, such as a crack. Results for both types of mesh are reported below. As would be expected, the irregular mesh results are not as good as those provided by a regular mesh with the same number of variables.

The problem is analyzed using various shell elements available in ABAQUS and different mesh densities. Thus, the example provides an indication of the relative efficiency of these elements.

Problem description

Results and discussion

Input files

References

Tables

Table 2.3.1–1 Shell roof: results for vertical displacement at the middle of the free edge, based on various regular meshes.

Element typeMeshVertical displacementError compared to 91.2 mm (3.59 in)
(mm) (in)
STRI34 × 467.442.665–25.8%
8 × 880.523.170–11.7%
18 × 1888.933.501–2.5%
S4R54 × 4109.604.31520.2%
8 × 895.993.7795.3%
18 × 1892.533.6431.5%
S4R4 × 4109.24.29819.7%
8 × 895.913.7765.2%
18 × 1892.613.6461.6%
S44 × 495.483.7594.7%
8 × 892.373.6371.3%
18 × 1891.893.6180.77%
S4R*4 × 4100.853.97110.6%
8 × 894.433.7183.5%
18 × 1892.883.6571.8%
S8R52 × 292.893.6571.9%
4 × 491.743.6120.6%
9 × 991.723.6110.6%
S8R2 × 289.173.511–2.2%
4 × 492.413.6381.3%
9 × 991.903.6180.8%
S9R52 × 292.893.6571.9%
4 × 491.743.6120.6%
9 × 991.723.6110.6%
SC6R4 × 466.422.615–27.1%
8 × 880.773.180–11.4%
18 × 1889.5863.527–1.75%
SC8R4 × 4110.94.36721.6%
8 × 896.803.8116.15%
18 × 1893.273.6722.28%
STRI652 × 274.672.940–18.1%
4 × 490.113.548–1.2%
9 × 991.673.6090.5%
S3R4 × 465.712.587–27.9%
8 × 880.113.154–12.1%
18 × 1888.903.500–2.5%
S3R**4 × 465.432.576–28.2%
8 × 880.373.164–11.9%
18 × 1890.683.57–0.6%
S3RS**4 × 467.872.672–25.6%
8 × 884.613.331–7.2%
18 × 1891.143.588–0.06%
*ABAQUS/Explicit element with enhanced hourglass control
**ABAQUS/Explicit element with default hourglass control

Table 2.3.1–2 Shell roof: results for vertical displacement at the middle of the free edge, based on irregular meshes.

Element typeMeshVertical displacementError compared to 91.2 mm (3.59 in)
(mm) (in)
STRI3coarse (258 d.o.f.)72.572.857–20.4%
fine (894 d.o.f.)83.343.281–8.6%
S4R5coarse (258 d.o.f.)96.573.8025.9%
fine (894 d.o.f.)93.983.7003.1%
S4Rcoarse (270 d.o.f.)96.163.7865.5%
fine (918 d.o.f.)93.933.6983.0%
S4coarse (270 d.o.f.)88.373.479–3.1%
fine (918 d.o.f.)91.943.6200.83%
S4R*coarse (270 d.o.f.)86.663.412–4.95%
fine (918 d.o.f.)93.113.6662.11%
S8R5coarse (270 d.o.f.)79.983.149–12.3%
fine (918 d.o.f.)91.033.584–0.2%
S8Rcoarse (210 d.o.f.)55.782.196–38.8%
fine (702 d.o.f.)89.643.529–1.7%
S9R5coarse (270 d.o.f.)82.753.258–9.2%
fine (918 d.o.f.)93.803.6932.9%
SC6Rcoarse (270 d.o.f.)71.02.796–22.1%
fine (918 d.o.f.)83.73.294–8.24%
SC8Rcoarse (270 d.o.f.)97.63.8437.05%
fine (918 d.o.f.)94.73.7303.90%
STRI65coarse (270 d.o.f.)81.533.209–10.5%
fine (918 d.o.f.)90.803.575–0.41%
S3Rcoarse (258 d.o.f.)70.562.778–22.6%
fine (894 d.o.f.)82.963.266–9.0%
S3R**coarse (258 d.o.f.)72.592.858–20.4%
fine (894 d.o.f.)84.283.318–7.6%
S3RS**coarse (258 d.o.f.)74.852.947–17.9%
fine (894 d.o.f.)89.333.517–2.0%
*ABAQUS/Explicit element with enhanced hourglass control
**ABAQUS/Explicit element with default hourglass control


Figures

Figure 2.3.1–1 Barrel vault roof problem.

Figure 2.3.1–2 Coarse irregular mesh for barrel vault.