Examples

Calculation #     . Wed May 29 14:15:37 PDT 2002

Example 1
Input:
Radius of sphere R: 1.0
Length of the cap L: 2.0
Poisson's ratio of the material ν: 0.3
Shell thickness h: 0.00633
Type of loading: 4
Number of buckles:: 1

This type of loading is often used in numerical analysis because it is simple to realize: at the first step the end shortening is prescribed (without rotation) until the axial load reaches the given value,
at the second step the probing lateral force is applied while the ends of the shell are fixed (a1=0, a2=0). In this case k=0 and the load deflection diagram as well as lower local buckling load
does not depend on number of buckles. It depends only on Batdorf parameter. The energy barrier is proportional to n. However, this type of loading rarely happens in practice.
Output:

Energy barrier diagram.

Batdorf parameter Z is 602.8052

KDF corresponding to lower local buckling load is 0.5113768 with energy barrier 0.0064962977 and deflection 4.17427
Buckling perturbation force Qmax is 0.24156956

KDF corresponding to upper local buckling load is 0.6293768 with energy barrier 0.0015018377 and deflection 1.7904705
Buckling perturbation force Qmax is 0.16387221

Axial load NDeflection wEnergy
barrier En		Force maxQ
0.7 1.2778592 6.5096945E-4 0.12220484
0.69 1.3407563 7.359705E-4 0.12800531
0.68 1.4067982 8.304434E-4 0.13383144
0.67 1.4749367 9.3551463E-4 0.13968706
0.66 1.5462202 0.0010524853 0.14557606
0.65 1.6216966 0.0011828594 0.15150347
0.64 1.7013663 0.001328379 0.15747449
0.63 1.7852291 0.0014910782 0.1634953
0.62 1.8743333 0.0016733475 0.16957204
0.61 1.9697272 0.0018780151 0.17571159
0.6 2.0724592 0.0021084545 0.18192177
0.59 2.182529 0.002368732 0.1882105
0.58 2.30413 0.0026638205 0.19458649
0.57 2.4372623 0.0029998885 0.20105942
0.56 2.587167 0.0033847692 0.20763965
0.55 2.7590857 0.0038287013 0.2143384
0.54 2.9603565 0.004345679 0.221168
0.53 3.2108965 0.004956228 0.22814238
0.52 3.5568304 0.0056948815 0.23527643
0.5113768 4.17427 0.0064962977 0.24156956

Example 2
Input:
Radius of sphere R: 1.0
Length of the cap L: 2.0
Poisson's ratio of the material ν: 0.3
Shell thickness h: 0.00633
Type of loading: 1
Number of buckles:: 1

Values of lower and upper local buckling loads are significantly less at this more common type of loading.

Output:

Energy barrier diagram.

Batdorf parameter Z is 602.8052

KDF corresponding to lower local buckling load is 0.47037682 with energy barrier 0.0107032275 and deflection 5.3462524
Buckling perturbation force Qmax is 0.2726345

KDF corresponding to upper local buckling load is 0.5203768 with energy barrier 0.005279257 and deflection 3.0945368
Buckling perturbation force Qmax is 0.23448403

Axial load NDeflection wEnergy
barrier En		Force maxQ
0.7 1.2684246 6.47165E-4 0.12215073
0.69 1.3302735 7.3116546E-4 0.1279441
0.68 1.3942188 8.2438433E-4 0.13376184
0.67 1.4602607 9.2788565E-4 0.13960776
0.66 1.5294476 0.001042887 0.14548597
0.65 1.6007309 0.001170779 0.15140142
0.64 1.6762074 0.0013131676 0.15735903
0.63 1.7548288 0.0014718977 0.1633649
0.62 1.8386916 0.0016491156 0.16942464
0.61 1.9267476 0.0018473087 0.17554502
0.6 2.0200448 0.0020693939 0.18173344
0.59 2.119632 0.0023188018 0.18799739
0.58 2.2255087 0.0025995937 0.19434534
0.57 2.34082 0.002916626 0.20078614
0.56 2.4645176 0.0032757372 0.20732962
0.55 2.5997465 0.0036840504 0.21398611
0.54 2.7486029 0.0041503585 0.22076707
0.53 2.9142318 0.004685706 0.22768492
0.52 3.1018748 0.00530428 0.23475331
0.51 3.3178215 0.00602487 0.24198729
0.5 3.5756996 0.0068734773 0.24940349
0.49 3.8985713 0.007888737 0.25702056
0.48 4.3514304 0.0091361245 0.2648595
0.47037682 5.3462524 0.0107032275 0.2726345