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Local structural stability evaluation using model reduction

Herein, a practical method is proposed to evaluate the local stability of large-size structures by combining two methods: direct evaluation and model reduction. The local stability of structures can be evaluated using a direct evaluation method that investigates the second variation of the local strain energy via nonlinear finite element (FE) analysis. For FE solutions with large degrees of freedom, the computational efficiency is improved significantly by applying the model reduction method. This combined method offers three primary advantages. First, it can be applied to any arbitrary part of a general structure. Second, it accounts for complicated interactions between local and global structures. Finally, it can be used easily by engineers in industries to reduce the computational cost. The effectiveness of the evaluation methodology is presented based on two numerical examples.


Introduction
Steel is typically used in civil, ship, and offshore structures.
Steel can be used to design thin-walled structures because it is characterized by high strength and toughness. However, these characteristics are likely to cause local structural failures, primarily due to buckling. Structural instability, such as buckling, is a very dangerous phenomenon in terms of structural safety.
Buckling refers to the stability loss of a component structure. The design standards for determining the resistance capacity of local structures involve the use of the design formula for buckling and ultimate strengths, which are specified in Codes and Rules [1]- [12]. Problems arise when the local structure is of a non-typical shape that is not predefined in the formula. In this case, the geometry shape and load pattern of the local structure are idealized to a pre-defined evaluation category in the design formula [2] [3]. However, this approach can significantly reduce the reliability of the formula. Further investigations into the arbitrary shape of local structures are essential to accurately predict the critical load of local buckling. To evaluate any arbitrary geometric shapes, a numerical evaluation method using the nonlinear finite element method, which can be applied to any shape without limitation, is typically recommended [13]. However, for an FE model with a large degree of freedom, considerable computational resources are required owing to the iterative solving involved to obtain nonlinear solutions.
As a cost-effective FE analysis procedure, Oh et al. [18] proposed an enhanced computational method using model reduction for large floating offshore structures with nonlinear structural behavior. The static condensation technique developed from the Guyan model reduction method was used [19]. The time required to compute forces on the local structures with nonlinear behavior was effectively reduced compared with that required in a nonlinear analysis involving a full FE model.
In this paper, a practical method is proposed to evaluate local buckling by combining two methods reported previously [17] [18].
The critical point at which the local structure loses its stability is explicitly determined, and its resistance capacity is quantified using the direct evaluation method. The computational efficiency is improved significantly by applying the model reduction method to an FE model with a significant number of degrees of freedom. The practicability of the proposed method is presented based on two examples: a horizontally stiffened cylindrical shell structure with radial bulkheads, and a bracket girder in the column-pontoon connection structure of the tension leg platform. The evaluation involves a nonlinear finite element analysis based on the Newton-Raphson method or the arc-length method [20].
The proposed method offers three primary advantages:  First, the method can be applied to any arbitrary part of general structures.
 Second, the complicated interactions between local and global structures ware fully considered without requiring local FE analysis.
 Finally, engineers in the industry can use this method easily to reduce computational costs.
In Section 2, structural stability is briefly reviewed, and the formulation used in the proposed method is presented. In Section 3, the performance of the proposed method is verified based on two numerical examples, and the computational efficiency of the model reduction is analyzed. Finally, the conclusions are presented in section 4.

Variation criterion for stable state
Structural stability refers to the phenomenon in which a structure returns to an equilibrium state when an external displacement disturbance is applied to the structure under this equilibrium state. To maintain a structurally stable state, the potential energy at the equilibrium state must be minimum [21].
An example of abrupt snap-buckling is introduced to assess the structural instability, as shown in Figure 2 In the equilibrium state, the potential energy of Π is always constant. If Π is a continuous derivative, then the function of Π may be expanded into a Taylor series with second-order variations in the equilibrium state, as follows: where E , ε , L , and A are the Young's modulus, strain, length, and section area of the truss member, respectively; α is the initial angle; θ is the applied angle; P is the applied load; q is the applied vertical displacement.
To determine whether the equilibrium state is stable, the secondorder derivative of Π is calculated as follows: The truss is stable if the expression above yields a positive number, i.e., Subsequently, the following condition is yielded: By adding rigidity to the structure exhibiting snap-through, the change in snap-through can be investigated. Figure 2(b) shows a case where a grounded spring is added to the load application point.
Considering the case of a spring in the direction of the load, the potential energy with vertical stiffness can be obtained as follows: where K is the additional grounded spring stiffness.
To determine the stability of the equilibrium states, the secondorder derivative of Π is calculated as follows: The truss is stable if the expression above yields a positive number, i.e., When the load is constant, the formulation of the second variation of the potential energy can be simplified to a second variation of the strain energy [16]. For practical purposes, it can be assumed that no load change occurs due to the small external displacement disturbances. When this assumption is applied, the term of the second-order work pertaining to the constant load ( P ) is absent in Equations (5) and (8). Therefore, when the second variation of the strain energy stored in the structures is from positive to negative, the structure loses stability. This is the energy criterion for evaluating the stability of structures. The structural resistance can be defined as the maximum load level within the stable load range obtained from the stability evaluation.

Direct evaluation procedure via FE analysis
The stability of the structure can be easily evaluated via nonlinear FE analysis. The energy criterion for the target structure is applied using the incremental strain energy value at each analysis step while incrementing the load or displacement. If the analysis is performed at sufficiently small intervals, then the instant-losing stability can be precisely determined. In the equilibrium state, the second variation of the strain energy becomes the second-order term of the incremental external work, based on the law of energy conservation. Depending on preference, the second-order term of the incremental external work can be evaluated as an energy criterion for stability evaluation.
where i δ f and i δu are the first variations of the force and displacement distributions along the boundary of the local structure, ( ) ( ) The local load ratio ( i λ  ) is calculated such that the error function ( i ϕ ) has a minimum value, and this is known as the approximated load method.
For comparison, another measure, i.e., the local load ratio ( i λ ), was used. It is defined as the ratio of the norm values (magnitudes) of the critical local load and reference local load as follows, and this is known as the norm value method: Because the translational and rotational degrees of freedom have different unit systems, it is inappropriate to calculate them simultaneously in Equations (12) and (13). Therefore, only the translational degrees of freedom of the nodal load components were considered in both methods to calculate the local load ratio.
The contribution of the rotational degrees of freedom to the strain energy was not considered, as they are negligible in most numerical problems. ( Step 3) The critical state, i.e., the stability limit, is determined.
The stability limit is determined by the load level at which the sign of the second variation of the strain energy shifts from positive to negative at each analysis step, as shown in Figure 4(c).
The highest local load ratio within a stable load range is selected as the critical local load ratio c λ  . The stability of the structure can be evaluated by the second variation of the strain energy, which is obtained from the second-order term of the incremental exter- The displacement index was introduced to investigate tendencies such as the severity of deformation at a specific load step.
The correlation curve between the local load ratio and the displacement index was derived from the strain energy at each step.
The scalar value of the incremental displacement index ( i x ∆ ) is calculated as follows: in which i λ ∆  is the incremental local load ratio.
(Step 4) The usage factor (η  ) is calculated as follows: in which s λ  is the load factor corresponding to the service load.
The local safety factor (LSF) with an inverse form of the usage factor was proposed as a measure of safety in the structural design. A higher safety factor implies a safer local structure.
The safety factor, characterized by the ratio of the load levels, is used to accurately evaluate the residual buckling strength of the local structure in a more rational manner.

Applicability of model reduction method
In general, the FE model of a global structure has many degrees of freedom, as compared with that of the local structure. In this case, from a computational cost perspective, it is useful to condense the stiffness for the degrees of freedom except for the region of interest using the model reduction method [18], and proceed to perform the nonlinear analysis only for the region of interest, as shown in Figure 5. A condensed stiffness is linear and does not require nonlinear iterations at each time; therefore, the computational cost can be reduced significantly.

Horizontally stiffened cylindrical structure with radial bulkheads
In the mono-column-type structure, a stringer structure is designed between the radial bulkhead and the cylindrical shell to resist the external pressure load, as shown in Figure 6. To investigate the behavior of the internal members of the cylindrical structure, a simplified and reduced-size model considering the local stringer area was established, as shown in Figure 7.    As shown in Figure 10, the nonlinear analysis was performed only on the stringer region of interest while model reduction was performed in the other regions. In addition, the analysis was performed by incrementing the prescribed displacement at the initial reference load application. Figure 11 shows an extended nonlinear analysis domain with an extended model reduction.     Table 1 and Figure 13. Considering the result of the full model analysis as a reference value, the difference in the model reduction result was calculated to be 8%; however, in the extended model reduction case, it was reduced to -3%. The difference in the prescribed displacement analysis result was calculated to be 16%; this was attributed to the prescribed displacement constraint of the local structure, which resulted in a greater stability as compared with the actual capacity. Based on comparing the results, it was found that the extended model reduction was the best method among those investigated.

Bracket girder in column-pontoon connection structure of tension leg platform
The local bracket girder structure in the tension leg platform (TLP) hull, which has a non-typical geometry that is not applicable to the conventional design formula, is presented as another example herein. The TLP is a vertically moored floating structure for deep water, as shown in Figure 14. The structural model for the column-pontoon connection evaluated in a previous study [17] was used, and further evaluation using the model reduction method was conducted in this study. The horizontal pontoon is typically connected to the vertical column, which is locally reinforced by a stiffener and a bracket girder, as shown in Figure 15.    Because the sign of the second variation of the strain energy was positive, the structure was stable until the external load was 3.5 times the reference (service) load, as shown in Figure 16. As shown by the local load ratio-displacement index curve, the highest local load ratio was 3.25 when the approximated load method was used. However, the result of stability verification indicated that the local load ratio of 3.22 was the stable state limit, which is the representative critical local load ratio, as shown in Figure 17. The first instability local load ratio based on the norm value method was 3.31, which is 2.8% higher than that obtained using the approximated load method. After the stable state limit, the local load ratio by the norm value method did not decrease significantly as compared with the approximated method.
The structural stability was evaluated by applying a model reduction method for the local bracket girder. As shown in Figure   18, nonlinear analysis was performed only on the region of interest in the bracket girder. Two analysis cases were investigated, namely, the model reduction of the remaining regions, and the prescribed displacement increment based on the reference displacement. Figure 19 shows the extended nonlinear analysis domain for the extended model reduction.    Table 2 and Figure   21. Considering the result of the full model analysis as a reference value, the difference in the model reduction result was calculated to be 12%; however, for the extended model reduction, it  reduced to -6%. The difference in the prescribed displacement analysis result was 4%, which was the lowest. However, an unrealistic result was obtained, where the structure became stable in the next step immediately after the first instability of the local structure occurred in the prescribed displacement analysis case.
It was found that the prescribed displacement analysis may yield unstable results, and that the calculated critical local load ratio was low compared with the maximum local load ratio shown in Figures 20 and 21. Therefore, the extended model reduction provided more reliable results among the reduced analysis cases. The computing efficiency was verified for this example, which involved a large degree of freedom. The required computing time was measured for each evaluation method, i.e., model reduction and prescribed displacement. The former evaluation method required a certain amount of time to perform static condensation. Table 2, the extended model reduction method provided the most reliable result among the reduced domain methods, except for the prescribed displacement case. In addition, the time required for the analysis of the local region reduced to 11.9% (= 240/2,023) compared with the full domain, as shown in Table 3 and Figure 22.

Conclusion
The local structure of ships and offshore structures has changed from a typical detailed design in the past to various shapes by location. Therefore, the area to evaluate the stability of the local structure has expanded and necessitates considerable computing resources.
By applying the proposed method, the instability of the local structure can be explicitly identified, and the local resistance capacity can be precisely quantified based on the LSF. For large FE models, the analysis time can be reduced using the model reduction method, which allows nonlinear iterative solving to be performed only for the local structure. Reliability and computational efficiency were investigated, and it was found that the extended model reduction method yielded the best solution.
In future studies, the proposed evaluation method will be applied to demonstrate the strength of the local structures of various arbitrarily shaped ships and offshore structures. In addition, how to determine the extended analysis region in the extended model reduction method will be further investigated to increase reliability.