Masahiko Fujikubo
4.4 Collapse analysis
4.4.2 Collapse analysis of VLFS in waves
Method of nonlinear structural analysis
In order to assess the structural safety of VLFS under abnormal environ- mental conditions, it is important to determine the ultimate failure mode of an entire structure and the corresponding ultimate capacity, by performing a progressive collapse analysis. The unit structure of a VLFS having I-type cross-section, as shown in Figure 4.3, has little reserved capacity after either a deck panel or a bottompanel have reached their ultimate strength. The load redistribution after the collapse of primary structural member is there- fore expected only in the planar directions. This again shows the importance of the progressive collapse analysis at an overall structural level.
As a result of the progress of computational capability, the nonlinear FE analysis is applied to the collapse analysis of structural systems, as exem- plified in Figures 4.10 and 4.11. Such a direct FE simulation, however, still cannot be applied to the entire VLFS having huge dimensions.
A simplified method for the progressive collapse analysis of a VLFS using the ISUM is presented in this section. ISUM, in its original version developed by Ueda (1984), is based on a matrix formulation similar to conventional FE analysis, but it employs particular definitions of elements, which are of the
same size and scale as the structural members themselves. Nonlinear buckling behavior of a structural unit (e.g. one plate panel between stiffeners) is ide- alized by appropriate shape functions assumed based on the collapse modes. Nonlinear contributions of post-buckling deflection to the membrane strains of plate panels are evaluated based on the theory of elastic, large deflection analysis. More details about ISUM formulation for the plate and stiffened plate models can be found in (Fujikubo 2002).
As a most fundamental case, the progressive collapse analysis of the unit structure taken from a floating airport VLFS model was performed, as shown in Figure 4.12 (Fujikubo 2003a). The deck and bottomstiffeners are mod- eled by beam-column elements, the plate panel between stiffeners by ISUM plate elements (one element for one bay), and the longitudinal and trans- verse bulkheads by Timoshenko beam elements. The number of elements is two orders of magnitude smaller than that of the conventional FE analysis. Figure 4.13 shows the calculated bending moment–curvature relationship for the case of longitudinal bending. The ultimate strength was attained when the deck or bottompanel on the compression side of bending reached their ultimate strength, and for further increase of applied curvatures the load-carrying capacity rapidly decreases.
Progressive collapse analysis of a pontoon-type VLFS in waves
By applying ISUM, the progressive collapse behavior of a pontoon-type VLFS in waves was analyzed (Fujikubo 2005). A floating airport model with two 4,000-mlong runways, as shown in Figure 4.14, was taken as an
Figure 4.13 Moment–curvature relationships of unit structure of VLFS. Source: Fujikubo 2003a.
Structural analysis and design of VLFS 83
Figure 4.14 Model of prototype floating airport.
Figure 4.15 Extreme response of transverse stress at bottom in Class-2 irregular waves (H1/3= 4.8 m, T1/3= 6.7 sec).
example structure. Multi-directional (short-crested) irregular waves from the transverse direction were considered. This wave direction gives the severest design condition with respect to the buckling strength of deck and bot- tomplating having a longitudinal stiffening system(Fujikubo 2003a). The following three-step approach was applied.
Step 1: The hydroelastic global response analysis is performed. The equation of motion with respect to the nodal displacement {d} can be expressed as [MS+ MA] ¨d + [N]˙d +KLS + KR {d} = {FW} (4.5)
with structural mass matrix[MS], added mass matrix [MA], hydrodynamic damping matrix [N], linear elastic stiffness matrix of structure KL
S
, hydrostatic restoring force matrix[KR] and wave exciting force {Fw}.
Step 2: Fromthe results of hydroelastic response analysis{de}, the external force vector{Fe} including the inertia and damping effects is calculated from
{Fe} = {FW} − [MS+ MA] ¨de − [N]˙de (4.6) where {Fe} gives the solution {de} of Eq. (4.5), when it is applied to the floating structure in a quasi-static manner as
KSL+ KR
{d} = {Fe} (4.7)
Step 3: The time history of the external force vector{Fe} is generated using the wave spectrumof irregular waves. It is then applied to the ISUM model in a quasi-static manner; that is the following nonlinear quasi-static equation is solved to obtain the progressive collapse behavior:
KSNL+ KR
{d} = {Fe} (4.8)
As described above, the progressive collapse analysis is performed by assum- ing that the dynamic external force distribution is the same as in the hydroelastic behavior and that only the structural stiffness changes as a result of the collapse. This assumption can be made when the collapsed area is limited as compared to the whole VLFS area (Fujikubo 2003b).
The Class-2 environmental condition for the ultimate strength limit state (see Table 4.1) was considered; that is, a 10,000-year return wave (100 times the number of service years) was estimated by the extreme wave statistics of Tokyo Bay. The significant wave height H1/3and wave period T1/3 are 4.8 mand 6.7 s, respectively.
The hydroelastic global response analysis at Step 1 was performed using the 3D detail method (Seto 2003) and the equivalent orthotropic plate model. Figure 4.15 shows the extreme response of the transverse membrane stress at the bottomplating due to bending, corresponding to the probability of exceedance of 1/1,000 in the short-term sea state. The maximum stress response is observed on the lee side and near the transverse edges.
Using ISUM, the computational time for the progressive collapse analysis at Step 3 can be significantly reduced when compared to the conventional FE analysis. It was found, however, that the VLFS (see Figure 4.14) was still too large to be solved directly. So, considering the elastic stress response shown in Figure 4.15, a 1,650 m × 495-marea shown by the dashed lines in Figure 4.14 was analyzed. In addition, the collapse behavior of all
Structural analysis and design of VLFS 85
Figure 4.16 Model for collapse analysis.
(a) (b)
Figure 4.17 Deflection and spread of collapse region (H1/3 = 4.8 m, T1/3 = 6.7 sec).
(a) Multi-directional irregular wave. (b) Uni-directional.
rectangular plate panels within the 19.8 m× 15 marea between two adja- cent longitudinal and transverse bulkheads was assumed to be the same. The model for the ISUM analysis is shown in Figure 4.16. The sandwich grillage model was employed to allow for the shift in the neutral axis due to buckling and yielding of the deck and bottompanels.
Figure 4.17(a) shows the deflection mode when the maximum deflection took place in the generated 2-h time history of deflection in multi-directional
irregular waves. None of the deck and bottomstiffened panels reached their ultimate strength in this case. For comparison purposes, the progressive collapse analysis in unidirectional irregular waves was performed, although it gives too severe a wave condition. In this case, the collapse occurred at the bottompanel with a smaller thickness than the deck panel as shown in Figure 4.17(b). It was found that the collapse area is confined to a relatively small area even in such an unrealistically severe wave condition.
The progressive collapse analysis, as shown above, is important not only for the prediction of the ultimate capacity of structure but also for the assessment of catastrophic failure scenario, its consequence, and the resulting risk.
4.5 Concluding remarks
Recent progress in linear and nonlinear structural analyses for the design of VLFSs has been presented. The major achievements can be summarized as a structural modeling technique that permits a rational analysis of VLFSs of huge structural sizes and a combination of the hydroelastic global response analysis with either the stress analysis of a detailed structure or the progres- sive collapse analysis of a global structure. Although not addressed in this chapter, significant progress has also been made in the probabilistic safety assessment of VLFS (Katsura 1999; Fujikubo 2003a).
Hitherto, studies on the structural analysis of VLFSs have been mainly concerned with a pontoon-type VLFS. More studies on structural modeling and progressive collapse analysis for semi-submersible-type VLFSs should be undertaken. There have been some attempts made on the structural opti- mization of a pontoon-type VLFS (Ma 1999; Yasuzawa 2003). The study on this area should be further pursued.