Imperfection

Furthermore, in the case of bifurcation problems, imperfection is sometimes helpful and even necessary since the solution might not be unique. Consider the Force (F) - Displacement (u) plot, shown in Figure 1. In the numerical simulation, the Finite Element mesh is usually based on perfect geometry, and the exact response cannot be captured very well. The numerical result (blue curve) shows that the limit point is overestimated. By considering a suitable and realistic imperfection, the secondary path (red curve) is achieved, which is physically correct. Therefore, imperfection helps to transform the bifurcation problem into a limit point problem, which can be dealt with the arc-length method or other numerical methods. This method is used for numerical investigations.

Figure 1. Stability Problem with Bifurcation Diagram

There are two methods to define the source of imperfection shape:

• Use of the results in the form of H3D file (TYPE=H3DRES) from an initial analysis

  The results of deformation or buckling modes (in case of a linear buckling analysis) from another analysis in the form of an H3D file is used with an amplitude/scaling factor to apply the initial imperfection. The H3D result file should exist already and should contain at least one subcase with displacement results or buckling modes.

• Apply the value of imperfection directly on the grid points (TYPE=GRID)

  Imperfection is defined directly on the GRID points based on the grid coordinate system in the IMPERF Bulk Data Entry.

The following input data are required to apply imperfection:

• ASSIGN I/O Options Entry
• IMPERF Subcase Information Entry
• IMPERF Bulk Data Entry

The first line of the IMPERF card is:

The second line of the IMPERF card is used to reference the source of imperfection, if the source is from an H3D result file:

If the imperfection is directly applied on the grid, then the TYPE field is set as GRID and X, Y, Z fields of the following continuation line define the imperfection:

Imperfection can be activated through the IMPERF Subcase Information Entry. In addition, a IMPERF Bulk Data Entry should specify the particular imperfection shape. Different subcases can contain different IMPERF entries. Imperfection can only be defined in the first subcase, if the subcase continuation is activated.

ASSIGN, H3DRES, 11, ./buckling_modes.h3d
SUBCASE 101
     Analysis = NLSTAT
     NLPARM(LGDISP) = 1
     IMPERF = 2
     ….
BULK DATA
IMPERF, 1, H3DRES
11, 100, 1, 1.0	     $$ H3D file with id 11, Subcase 100, mode 1, scaling factor 1.0
11, 100, 2, 0.5	     $$ H3D file with id 11, Subcase 100, mode 2, scaling factor 0.5
11, 100, 3, 0.1	     $$ H3D file with id 11, Subcase 100, mode 3, scaling factor 0.1

The lower end (A) is fixed, while the other nodes are constrained along the Y-translational; X and Z rotational degrees of freedom. A linear elastic material model with an elastic modulus of 211 GPa and Poisson’s ratio of 0.312 was considered.

Figure 2. Model Details and First Buckling Mode of the Structure

The problem was solved with the arc-length method and the variation of load with lateral displacement for different values of imperfection is:

Figure 3. Load vs. Lateral Displacement Plots. for different values of imperfection

A single point load is applied on a curved shell of thickness 6.35 mm, which has two opposite edges constrained along all three translational degrees of freedom (Figure 4). The shell material has an elastic modulus of 3.102 GPa and Poisson’s ratio of 0.3. Linear buckling analysis was performed under these conditions and the buckling mode shapes were extracted.

Figure 4. Model with Boundary Conditions

The arc-length method was used to solve the problem, with and without imperfection. In the analysis with imperfection, the first buckling mode was used for the imperfection, with a scaling factor of 1% of shell thickness (0.0635).

Figure 5. 1st Buckling Mode

From the load vs. displacement plots, the curve with imperfection exhibits a much lower limit point compared to the curve without imperfection. The scaling factor of imperfection is not sensitive for this example; thus, not investigated further.

Figure 6. Load-Displacement Curve of the Cylindrical Roof from OptiStruct

1. Imperfection is supported for nonlinear static and nonlinear direct transient for both small and large displacement analyses.
2. In the presence of imperfection, element checking is still based on the perfect mesh.
3. The magnitude of imperfection is not checked internally. It remains your responsibility to specify an imperfection that is physically correct.
4. Initial stress condition can be specified independently of the imperfection.
5. Multiple imperfections can be specified for different subcases in the same input file.
6. In the case of subcase continuation (CNTNLSUB), imperfection can only be specified in the first subcase. In other cases, it would lead to an error.
7. The output of displacement in the results files (for example, H3D) does not contain the value of imperfection. The output of displacement and strain is based on the imperfect configuration.
8. The imperfection shape and scaling factors depend on the structural boundary conditions and the type of loading. The suitable combination of buckling modes and their scaling factor must be chosen according to your requirement, which could be from experimental reference. The imperfection must be large enough to capture the buckling behavior but must also be suitably small enough to guarantee the accuracy of the results.