Reliability Investigation of a Shell Structure
A cylindrical panel was considered, which is mentioned e. g. in Kratzig (1989) and Schorling and Bucher (1999). The assumed structure is shown in Figure 8. The geometrical and the material proberties were given as: radius R = 83.33 m, the half width and heigth a = 5 m, the thickness h = 0.1 m, the Young’s modulus E = 3.4 x 1010 N/m2, the mass density p = 3400 kg/m3 and the Poisson’s ratio p = 0.2. The constant load factor is P = 1000 N/m.
Fig. 8. Nonlinear cylindrical shell structure with associated weighted imperfection shapes.
The structure is discretized with 7 x 7 nodes and meshed with geometrically nonlinear 9-node shell elements. At a static load factor of vcrit = 16825 the structure reaches an unstable state (Kratzig, 1989: vcrit = 15120; Schorling and Bucher, 1999: vcrit = 16200). The static load is assumed to be P0 = 0.85vcritP. The fluctuating load is considered as Pfluct = if(t)P, where f(t) is the unit white noise process and і is the load factor. The damping is assumed as modal damping with the damping ratio Zk = 0.02 for all modes.
The geometrical imperfections are considered in terms of radial deviations from the perfect panel surface and are modelled as a conditional Gaussian random field. The mean is assumed as zero and the standard deviation as a = 10-3 m. The correlation length of the exponential correlation function is considered with lH = 10 m. The imperfection shapes are obtained by the decomposition of the covariance matrix according to Equation (14). The first four imperfection shapes are shown in Figure 8 as well. The corresponding standard deviations aYi in uncorrelated normal space are indicated in the figure. The first shape is very similar to the buckling shape.
The structure was investigated by using the Ito analysis and it was found that only the first imperfection shape has a major influence on the stability behaviour. The critical noise intensity of the perfect system was obtained as D0crit = 92000я with the linear and D0crit = 20000я with the nonlinear method by averaging 20 simulations with 105 time steps. The nonlinear analysis uses a modal subspace spanned by 12 of the 213 eigenmodes with a critical time step of At = 6.3 • 10-3 s. The investigation of the first imperfection shape obtained by nonlinear analysis show observable deviations from the linear results. This points out that the nonlinearities of this structure have a higher influence as compared to the previous example. The obtained stability boundaries depending on the imperfection size are displayed in Figure 9 for the linear and the nonlinear analysis.
The failure probability for this one dimensional problem can be obtained analytical from the stability boundaries and is shown in Figure 10 depending on the noise intensity for both methods. It
Table 2. Critical static and dynamic loads.
is to be seen in the picture, that a sufficient approximation of the nonlinear probability graph is not possible with the linear method.
Furthermore the discretisation influence on the stability boundaries was investigated on the perfect panel. Additional to the 7 x 7 node model, systems modeled with 11 x 13 (Schorling and Bucher, 1999) and 25 x 25 nodes and meshed with geometrically nonlinear 9-node shell elements were analyzed. The critical static buckling loads are shown in Table 2. The dynamic stability boundaries are obtained by using the linear Ito analysis under considering the static load first with 0.85vcrit of the same model. This leads to different static loads. To obtain the stability boundaries by a constant static load this load was assumed as 0.85vcrit of the 7 x 7 node model. The results are shown additional in Table 2. It is to be seen that the influence of the discretization on the critical noise intensity is much higher than on the static buckling load.
The nonlinear method was not applicable for the 11 x 13 and 25 x 25 node models, caused by the hugh numerical effort. Simulations with the 11 x 13 node model by using a modal reduction from 657 eigenmodes to 33 eigenmodes (the critical time step is then 3.4 • 10-3 s) did not lead to sufficient results. The Lyapunov exponent did not converge to a stationary value, caused be the to short time window of the simulations, limited by the available computer capacities.
Arnold, L. and P. Imkeller (1994). Furstenberg-Khasminskii formulas for Lyapunov exponents via anticipative calculus. Technical Report Report Nr. 317, Institut fur dynamische Systeme, University of Bremen.
Bathe, K.-J. (1996). Finite Element Procedures. Englewood Cliffs: Prentice Hall.
Brenner, C. E. and C. Bucher (1995). A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading. Probabilistic Engineering Mechanics 10, 265-273.
Bucher, C. (2001). Stabilization of explicit time integration by modal reduction. In W. A. Wall, K.-U. Bletzinger, and K. Schweizerhof (Eds.), Proceedings, Trends in Computational Mechancics. Barcelona: CINME.
Bucher, C., O. Huth, and M. Macke (2003). Accuracy of system identification in the presence of random fields. In A. DerKiureghian, S. Madanat, and J. Pestana (Eds.), Applications of Statistics and Probability in Civil Engineering, pp. 427-433. Millpress.
Ditlevsen, O. (1991). Random field interpolation between point by point measures properties. In Proceedings of 1. Int. Conference on Computational Stochastic Mechanics, pp. 801-812. Computational Mechanics Publications.
Ghanem, R. and P. D. Spanos (1991). Stochastic Finite Elements – A Spectral Approach. New York/Berlin/Heidelberg: Springer.
Kratzig, W. (1989). Eine einheitliche statische und dynamische Stabilitatstheorie fur Pfad- verfolgungsalgorithmen in der numerischen Festkorpermechanik. Z. angew. Math. Mech. 69-7, 203-213.
Lin, Y.-K. and G.-Q. Cai (1995). Probabilistic Structural Dynamics. New York: McGraw-Hill. Macke, M. and C. Bucher (2000). Conditional random fields for finite elements based on dynamic response. In M. Deville and R. Owens (Eds.), Proc. 16th IMACS World Congress on Scientific Computation, Applied Mathematics and Simulation, Lausanne, Switzerland, August 21-25, 2000.
Matthies, H. G. and C. Bucher (1999). Finite Elements for Stochastic Media Problems. Comput. Methods Appl. Mech. Engrg. 168, 3-17.
Matthies, H. G., C. E. Brenner, C. G. Bucher, and C. G. Soares (1997). Uncertainties in Probabilistic Numerical Analysis of Structures and Solids – Stochastic Finite Elements. Struct. Safety 19, 283-336.
Most, T., C. Bucher, and Y. Schorling (2004). Dynamic stability analysis of nonlinear structures with geometrical imperfections under random loading. Journal of Sound and Vibration 1-2(276), 381-400.
Schorling, Y. and C. Bucher (1999). Stochastic stability of structures with random imperfections. In B. F. J. Spencer and E. A. Johnson (Eds.), Stochastic Structural Dynamics, pp. 343-348. Rotterdam/Brookfield: Balkema.
Soong, T.-T. and M. Grigoriu (1992). Random Vibrations of Mechanical and Structural Systems. Englewood Cliffs: Prentice Hall.