Analyzing a structure during lifetime close to reality requires to consider the complete load and modification process. The load and modification process of a structure is comprehended as the sequence of different states of construction, changes in material, e.g. the change of material behavior due to physical or chemical processes, structural alteration resulting from e.g. refurbishing or strengthening, and changes in load.

Due to the predominantly informal and lexical fuzziness of the parameters of the load and modification process the uncertain parameters are described by the mathematical model fuzziness. The time dependent parameters are considered as fuzzy functions

or fuzzy processes

 The coefficients indicate the time , the spatial coordinates , and further parameters , e.g. temperature.

 Figure 1: Fuzzy process

The particular fuzzy processes resp. fuzzy functions constitute a system of parallel and/or sequent functions. In the numerical simulation of the load and modification process the time discretized fuzzy input values are mapped onto the fuzzy result values. The mapping model represents an uncertain or crisp analysis algorithm. The numerical simulation is carried out with aid of the a-level-optimization. Any arbitrary deterministic analysis algorithm can be applied as fundamental solution of the mapping model. The choice of the deterministic fundamental solution depends on the type of structure to investigate. Appropriate deterministic fundamental solutions are e.g. FE-methods or the solution of a differential equation system.

Example: Natural stone arch bridge

 Figure 2: Draft of the bridge

The geometrically and physically nonlinear analysis of plane reinforced concrete, prestressed concrete, and steel bar structures is chosen as deterministic fundamental solution to analyze the stone arch bridge. The free span of the arch is 94 m, the arch rise amounts 17.50 m. A sophisticated statically model considers the elastic foundation as well as the nonlinear elastic stiffening effect of the back-up masonry. The strengthening and rehabilitation of the bridge by a masonry grouting is considered in the load and modification process.

 Figure 3: Static model

The load and modification process comprehends the loading of the bridge, the relieving, the strengthening of the masonry by grouting and the reloading.

Fuzzy input functions are

• the uncertain nonlinear spring stiffness
• the uncertain nonlinear and time dependent masonry behavior (strength, modulus of elasticity) before and after grouting
• the uncertain load.

 Figure 4: Interaction between the uncertain masonry strengths before and after rehabilitation

 As selected fuzzy result the uncertain process of the vertical displacement of the arch crown is shown.

Figure 5: Fuzzy result process - vertical displacement of the crown of the arch

The condition of a structure during the load and modification processes is assessed by structural indicators. Typical indicators are e.g.

• damage indicator resp. health indicator: change of eigenfrequencies or eigenvalues of the stiffness matrix K
• sensitivity: quantity of change of a result parameter caused by a determined load
• energy: Shannons’s energy.