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4 Verification of the structural elements

The evaluation of the structure by means of CSFM is carried out through two different analyzes: one for combinations of serviceability and one for combinations of the ultimate limit state. The usability analysis assumes that the limit behavior of the element is sufficient and the flow conditions of the material are not achieved at the usability level. This approach enables the use of simplified material models (with a linear branch of the stress-strain diagram of the concrete) for the serviceability analysis in order to improve the numerical stability and the calculation speed. Therefore, it is recommended to use the workflow presented below, where the first step is the analysis of the ultimate limit state.

4.1 ULT analysis

The assessment of the various verifications required for specific design standards is based on the direct results of the model. ULS verifications are carried out for the concrete strength, reinforcement strength and anchoring (composite shear stresses).

To ensure that a structural element is efficiently designed, it is highly recommended that a preliminary analysis be performed, taking into account the following steps:

• Make selection of the most critical load combinations
• Calculate only load combinations for the ultimate limit state (ULS)
• Use a coarse mesh (increasing the standard mesh size multiplier, Figure 23

\ [\ textsf {\ textit {\ footnotesize {Fig. 23 \ qquad Mesh multiplier.}}} \]

Such a model can be calculated very quickly so that designers can efficiently review the detailing of the structural element and rerun the analysis until all of the review requirements for the most critical load combinations are met. If all the requirements for checking this preliminary analysis are met, it is recommended to include the complete limit load combinations and the use of a fine mesh size (mesh size recommended by the program). The user can change the network size using the multiplier (value from 0.5 to 5).

The basic results and checks (stress, strain and utilization (i.e. the value / limit value calculated from the standard) as well as the direction of the main stresses in concrete elements) are displayed using various diagrams in which the pressure areas are usually shown in red and tension areas in blue .

Global minimum and maximum values ​​for the entire structure and minimum and maximum values ​​for each user-defined part can be highlighted. Advanced results such as tensor values, deformations of the structure and reinforcement proportions (effective and geometric), which are used to calculate the tensile stiffening of the reinforcing bars, can be displayed in a separate tab. In addition, loads and support reactions for selected combinations or load cases can be displayed.

4.2 SLS analysis

SLS evaluations are carried out for stress limitation, crack width and deflection limits.

The stresses in concrete and reinforcement elements are verified according to the applicable standard in a manner similar to those specified in the ULS.

The serviceability analysis contains certain simplifications of the material models that are used for the final ULS analysis. A perfect bond is assumed, i.e. the anchorage length is not checked for serviceability. Furthermore, the plastic branch of the stress-strain curve of concrete under compression is not taken into account, while the elastic branch is linear and infinite. These simplifications improve the numerical stability and the calculation speed, but at the same time do not reduce the generality of the solution as long as the resulting limit values ​​for the material stress when the usability is reached are well below their yield limits (as required by standards). Therefore, the simplified models used for usability are only valid if all requirements for verification are met.

4.2.1 Calculation of the crack width

There are two ways to calculate crack widths: stabilized and unstabilized cracks. According to the geometric reinforcement proportion in each part of the structure, a decision is made as to which type of crack calculation model is used (TCM for stabilized crack models and POM for non-stabilized crack models).

\ (\ textsf {\ textit {\ footnotesize {Fig. 24 \ qquad Crack width calculation: (a) considered crack kinematics; (b) projection of crack kinematics into the principal}}} \) \ (\ textsf {\ textit { \ footnotesize {directions of the reinforcing bar; (c) crack width in the direction of the reinforcing bar for stabilized cracking; (d) cases with}}} \) \ (\ textsf {\ textit {\ footnotesize {local non-stabilized cracking regardless of the reinforcement amount; (e) crack width in the direction of the reinforcing bar}}} \) \ (\ textsf {\ textit {\ footnotesize {for non-stabilized cracking.}}} \)

While the CSFM provides a direct result for most of the checks (e.g. component capacity, deflections ...), the results of the crack width are calculated from the results of the reinforcement strain, which are directly carried out by the FE analysis according to the methodology described in Fig. 24, to be provided. A crack kinematics without slippage (pure crack opening) is considered (Fig. 24a), which agrees with the main assumptions of the model. The main directions of stresses and strains define the inclination of the cracks (θr = θs = θe). According to Fig.24b, the crack width (w) in the direction of the reinforcement bar (wb), which leads to the following definition:

\ [w = \ frac {w_b} {\ cos \ left (θ_r + θ_b - \ frac {π} {2} \ right)} \]

with θb as rod inclination.

The calculation of the component wb is carried out consistently, based on the tensile stiffening models presented in Section 1.2.4, by integrating the reinforcement strains. For these areas with fully developed crack patterns, the calculated average elongations (em) along the reinforcement bars directly along the crack spacing (sr) integrated. While this approach for calculating the crack directions does not correspond to the actual position of the cracks, it nevertheless provides representative values ​​that lead to results for the crack width that can be compared with the values ​​required by the standard at the position of the reinforcing bar.

Special situations are observed at concave corners of the calculated structure. In this case, the corner defines the location of a single crack that does not behave in a stabilized manner before additional neighboring cracks develop. These additional cracks usually arise according to the usability area (Mata-Falcón 2015), which justifies the calculation of the crack widths in such an area as if they were not stabilized (Fig. 25), using the information presented in Section 1.2.4 Model.

\ [\ textsf {\ textit {\ footnotesize {Fig. 25 \ qquad Definition of the region at concave corners in which the crack width is computed as if it were non-stabilized.}}} \]