2.3.3 The pinched sphere problem

Products: ABAQUS/Standard  ABAQUS/Explicit  

This problem is chosen to provide verification and illustration of the axisymmetric shell elements in ABAQUS. Most of the response is localized, so the case represents a more severe test than, for example, a sphere with internal pressure. Koiter (1963) has provided an analytical solution, which has been used as a standard test for several axisymmetric shell finite elements (see Ashwell and Gallagher, 1976).

Problem description

Results and discussion

Input files

References

Table

Table 2.3.3–1 Displacement at top of sphere.

ElementNumber of elementsNormalized displacementError
type
SAX1, ABAQUS/Standard1015.52–24.7%
SAX1, ABAQUS/Standard2019.85–3.5%
SAX1, ABAQUS/Explicit1024.77–20.2%
SAX1, ABAQUS/Explicit2024.63–19.6%
SAX2515.62–24%
SAX21020.12–2.2%
S4R2019.80–3.9%
SC8R2019.83–3.7%
SC8R*2019.94–3.2%
The normalized displacement is , where w is the actual displacement; E is Young's modulus; t is the shell thickness; and P is the applied load.
Koiter's (1963) exact solution gives a normalized displacement of 20.6.
*ABAQUS/Standard results with enhanced hourglass control.


Figures

Figure 2.3.3–1 Pinched sphere example.

Figure 2.3.3–2 Radial displacement versus angle measured from the point of load application.

Figure 2.3.3–3 Energy balance for 20-element model, ABAQUS/Explicit analysis.