Numerical example – cylindrical shell strip (DSG)

In this section, the quality of the modified isogeometric NURBS shell formulations will be investigated for the cylindrical shell strip with a constant transverse line loadˆqx

analyzed before in Section 5.4.4.

In Figure 6.6, the effect of varying slenderness ^R_t on the discrete displacement results u_x^h_P of shear-rigid Kirchhoff-Love shell models is analyzed. The analytical beam solu-tion is compared to the results of both a pure displacement formulasolu-tion (3p) and the DSG ansatz (3p-DSG) with modified membrane strain componentsε^3p,h,DSG_αβ . Whereas the “3p” NURBS model shows significant membrane locking as already determined in Section 5.4.3, the results of the “3p-DSG” shell element are completely free from spuri-ous membrane strains, independent of the slenderness, and thus represent a locking-free shell formulation.

Beam reference3p-DSG3p

Slenderness ^R_t DisplacementuxP

10000 1000

100 10

1 0.8 0.6 0.4 0.2 0

Figure 6.6:Cylindrical shell – displacement convergence, 3p shells, DSG.

In addition to the Kirchhoff-Love shell formulations, in Figure 6.7, several shear de-formable 5-parameter Reissner-Mindlin shell elements are compared to the reference solution and among each other for the given problem setup. For the standard shell formulation (5p-stand.) of Section 4.4.1 with a difference vector imposed on the unde-formed normal, both transverse shear and membrane locking effects are assumed. The

hierarchic difference vector model (5p-hier.) of Section 4.4.2 is free from parasitic trans-verse shear effects, but is prone to membrane locking in the same way as determined for the Kirchhoff-Love (3p) model in Figure 6.6. In addition, the DSG versions of the two aforementioned 5-parameter shell models are examined, which are denoted as “5p-stand.-DSG” and “5p-hier.-DSG”.

Beam reference5p-stand.-DSG5p-hier.-DSG5p-stand.5p-hier.

Slenderness ^R_t DisplacementuxP

10000 1000

100 10

1 0.8 0.6 0.4 0.2 0

Figure 6.7:Cylindrical shell – displacement convergence, 5p shells, DSG.

Figure 6.7 numerically confirms the previous assumptions. The shell element with stan-dard difference vector (5p-stand.) shows the worst behavior as both unphysical trans-verse shear and membrane normal stresses deteriorate the accuracy. Looking at the DSG version of the standard model (5p-stand.-DSG), it turns out that for this example, removal of unphysical membrane contributions produces significantly better results than removal of transverse shear locking with the “5p-hier.” shell. Nonetheless, “5p-stand.-DSG” still exhibits transverse shear locking, which becomes particularly pronounced for a slenderness of^R_t >1000. The small difference between “5p-stand.” and “5p-hier.” re-sults from the predominant influence of membrane locking so that the additional effect of transverse shear does not become as obvious if both locking effects show up simulta-neously. Finally, the performance of the hierarchic isogeometric NURBS shell element with DSG modification (5p-hier.-DSG) is evaluated. The formulation is completely free from geometric locking and matches the reference beam solution very well, independent of the chosen slenderness. The DSG approach of Section 6.1.2, therefore, successfully removes the undesired parasitic strain contributions for the case of higher-order and higher-continuity NURBS shell discretizations.

In Table 6.1, the numerical displacement results of the entire family of NURBS shell element formulations with DSG modification of the in-plane strain components are

dis-played. In addition to the 3- and 5-parameter models, the three-dimensional 7-parameter shells with DSG ansatz (7p-stand.-DSG, 7p-hier.-DSG) are also shown. Whereas “7p-hier.-DSG” is again completely locking-free, the “7p-stand.-DSG” shell element is prone to both transverse shear and curvature thickness locking, which was already investigated in Section 5.4.4. The effect of curvature thickness locking on the standard difference vec-tor formulation can also be determined in Table 6.1 if the results of the “5p-stand.-DSG”

shell element are compared to the “7p-stand.-DSG” version particularly for a large slen-derness ^R_t.

Slenderness^R_t 10 100 1000 10000

Shell formulation (2nd order NURBS)

3p-DSG 0.9406 0.9444 0.9445 0.9445

5p-stand.-DSG 0.9396 0.9048 0.7560 0.2652

5p-hier.-DSG 0.9422 0.9445 0.9445 0.9445

7p-stand.-DSG 0.9403 0.9031 0.7158 0.0895

7p-hier.-DSG 0.9445 0.9445 0.9445 0.9445

Analytic result

Beam reference 0.9451 0.9425 0.9425 0.9425

Table 6.1:Cylindrical shell – displacementsu_xP, overview of shell formulations, DSG.

So far, only the dependency of the discrete radial displacementu_x^h_P on the slenderness of the structure^R_t has been analyzed for the cylindrical shell example. Therefore, in the following, the quality of the stress resultant bending momentmxx will also be investi-gated for the 5-parameter Reissner-Mindlin shell element formulations studied before in Figure 6.7.

A fixed slenderness of^R_t = 1000is selected, for which several shell models, according to Figure 6.7, already exhibit significant locking in the investigated discrete displacements.

For the computation of the bending momentmxx, a local Cartesian coordinate system is defined along the path P-Q as displayed in Figure 5.16, with thex- andz-axes to be tangential to the parametricθ¹- andθ³-directions, respectively. The curvilinear stress components σ^kl are transformed into the local Cartesian basisei according to Equa-tion (5.11).

With the local Cartesian stress componentsσ^xx at the two Gauss points inζ-direction, a linearly varying distribution of in-plane normal stresses through the thickness of the shell body is defined, which is subsequently multiplied with the thickness coordinate and then integrated inθ³-direction of the shell.

In Figure 6.8, the bending momentmxx, which was computed from the linear part of the in-plane normal stresses, is plotted along the path P-Q. The results confirm the observations and conclusions made for the numerical displacement evaluations.

Both the standard and the hierarchic pure displacement models are very sensitive to lock-ing for the defined slenderness. The bendlock-ing momentsmxxare approximately zero along

Beam reference5p-stand.-DSG5p-hier.-DSG5p-stand.5p-hier.

PathP−Q(arc length)

Bendingmomentmxx

16 14 12 10 8 6 4 2 0 4.0E-007 0.0E+000 -4.0E-007 -8.0E-007 -1.2E-006 -1.6E-006 -2.0E-006

Figure 6.8:Cylindrical shell – bending moments mxx, 5p shells, DSG.

the entire path P-Q. A modification of the membrane strains with the DSG method gen-erates significantly better results in the standard model (5p-stand.-DSG), which again reveals that the dominant locking phenomenon in this numerical test example is mem-brane locking. Still, however, transverse shear locking compromises the solution quality.

As for the discrete displacement tests, the bending moment of “5p-hier.-DSG” again conforms very well with the beam reference solution derived in Figure 5.17.

In this section, the DSG approach ofBletzinger et al.(2000) was successfully trans-ferred to higher-order and higher-continuity NURBS discretizations in order to remove membrane and in-plane shear locking. The higher continuity of the NURBS basis func-tions may, however, result in a coupling of degrees of freedom, which compromises the computational efficiency. For the 3-parameter and the hierarchic isogeometric shell ele-ments of this thesis, only the membrane part of the strains has to be modified to provide element formulations, which are completely free from geometric locking.