The process of developing a solution strategy

A solution strategy for NLFEA comprises choices regarding kinematic compatibility, material models and equilibrium, as illustrated in Tab. 6. Most analysis software include several possibilities for combining different material models, for using a variety of different element types and for solving the equilibrium equations with different iterative methods. A broad suite of methods or models should be available for the user, since most models will be more suited to some applications than others. However, the user should select the solution strategy with care

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and be aware of the consequences of the choices that are made (Vecchio 2001, fib 2008, Hendriks et al. 2017a). For example, the boundary conditions can play a significant role, either by introducing artificial tensile forces, leading to premature cracking, or by restraining longitudinal elongation, leading to an overestimation of the benefit from the compressive membrane effect in slabs and beams. In shell structures like the dam shown in Fig. 2, shell elements can seem to be the most natural choice of element type, however the inherent assumption of plane sections is most likely violated by cross-sectional thicknesses of several meters, leaving solid elements as the only viable option. Also, the application of loads, such as distributed pressures, will generally become more accurate for solid elements than for shell elements.

Tab. 6: Examples of the content of a solution strategy for NLFEA.

Kinematic compatibility

Finite element types for concrete and reinforcement, including order of numerical integration.

Finite element sizes.

Idealization of geometry.

Idealization of boundary conditions.

Material models

Material models for concrete and reinforcement.

Material models for possible interfaces and boundary conditions.

Equilibrium

Iterative methods for the solution of the non-linear equilibrium equations.

Convergence criteria and suitable tolerances.

Method for determining if the capacity was reached or not.

The engineer is left with two general approaches for selecting a solution strategy, either develop his or her own solution strategy for a specific purpose, or select a solution strategy based on a set of guidelines developed for safe use of NLFEA (Hendriks et al. 2017a, 2017b). The process of developing a solution strategy consists of the four main activities 1) definition, 2) verification, 3) validation and 4) demonstration of applicability, as illustrated in Tab. 7.

In the definition activity, the engineer would usually explore the available options in available software or review relevant literature. The results obtained from analyses with different solution strategies might be compared in order to find the most suitable for the purpose. The engineer can define the solution strategy following two different categories or philosophies, either

i) use sub-models with a range of free parameters that can be fitted to experimental observations, or

ii) select sub-models that are only functions of readily observable basic variables.

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An example of the first category can be if Eq. (2) was used as a sub-%'$'*, '-&6+

modulus without fixing the parameters of the model to the values predefined by the design code.

An example of the second category could be the solution strategy developed in the present work, where the compressive cylinder strength of concrete is the only required input to the material model for concrete. Note that which category that is selected by the engineer is a matter of taste and depends on whether experimental results are available for calibration or not. Many free parameters most likely leads to overfitting of the model (Beck & Yuen 2004), giving good predictions for the cases that were used in the calibration process, but most likely leading to worse predictions elsewhere. However, in a design situation or when an existing structure is being assessed, calibration is less applicable since no observations are usually available describing the true behaviour of the structure.

Tab. 7: The activities in the process of developing a solution strategy for NLFEA.

Definition Select suitable material models, element types, iteration methods, etc.

Verification

Apply fundamental checks to assess if the model works as expected and assess the sensitivity to variations of the solution strategy, e.g. mesh size sensitivity and load step size sensitivity.

Validation

Assess how well the NLFEA predictions compare to the real structural behaviour, i.e. quantifying the modelling uncertainty by comparing NLFEA predictions to experimentally obtained results.

Demonstration of applicability

Prove that the solution strategy is suitable for the intended purpose.

During verification and validation, the engineer would seek answers to the questions Are we solving the equations right? and Are we solving the right equations? (Roache 1998, Engen et al. 2017a). In the NLFEA context, verification thus comprises sensitivity studies of for example the element sizes, the element types, the load step sizes, the convergence tolerances and the iteration methods. Verification also comprise typical single element tests that are performed in order to check if the material models behave as expected. Validation on the other hand, is related to the idealization of the structural geometry, boundary conditions and material behaviour and thus involves quantification of the modelling uncertainty for the selected solution strategy. If a solution strategy was defined following category i) above, these activities would also include estimating the values for the free parameters of the sub-models. Based on the findings in the verification and validation activities, the engineer might take a step back to the definition activity and make changes to the solution strategy.

The last step involves a demonstration of applicability by testing the solution strategy on realistic cases similar to the practical design problem it is intended for. The purpose of the final activity is to reveal if the expected important phenomena can be captured and the level of detail in the results is sufficient to be used as a basis for decisions. Note that the process has a subjective nature, leaving to the engineer to decide what is good enough or detailed enough.

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The process of developing a solution strategy is further discussed in Paper II and III (Engen et al. 2015, 2017b) appended to the thesis.