# Realistic bending stiffness of diaphragm walls for structural analysis

### A comparison with the uncracked and totally cracked stiffness for the case of The Waalbrug Nijmegen

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## Abstract

For a safe structure design standards mainly emphasize on calculations based on the uncracked and fully cracked stiffness of concrete structures. In the common design practice an uncracked stiffness (EI0) is applied for the SLS and a fully cracked stiffness (EI?) for the ULS, where the reduced stiffness EI? ? 1/3EI0. However, in reality it is unlikely for a structure to remain uncracked or to become fully cracked; it is more common to find members consisting of cracked and uncracked zones. This implies the existence of a realistic variable bending stiffness (EIvar) along the reinforced concrete member. The Eurocode 2 states that members which are expected to crack, but may not be fully cracked, will behave in a manner intermediate between the uncracked and fully cracked conditions. Insufficient elaboration on the impact of EIvar on the safety of concrete structures and whether the applied design models using EI0 and EI? are indeed conservative models, led to the formulation of this research project. By means of the ‘Waalbrug-project’, this thesis investigates the impact of a realistic bending stiffness on the structural behaviour. For this particular project the safety of the packing structure, consisting of diaphragm walls and a roof structure, is analyzed. The bending stiffness of diaphragm walls is not constant over the height, but it varies as a function of the magnitude of the occurring bending moment and the amount of reinforcement. As soon as the wall is cracked the wall stiffness decreases at an increasing bending moment, which is explained in literature by means of the M-(N)-? diagram. A reduced wall stiffness results in greater wall deformations, but on the other hand the deformations on their turn influence the wall stiffness. For the influence of the soil behaviour on the stiffness of the diaphragm wall the ‘interaction’ model Plaxis 2D was used. Different calculation models were set up to determine whether the structural behaviour at EIvar lies indeed within the behaviour of an uncracked and fully cracked structure. The impact of the boundary condition (hinged or clamped connection) was also studied. The Half Model and Total Model in the calculation programs PCSheetPileWall and Plaxis 2D were used for structural analysis. The structural behaviour was expressed in terms of the bending moment (MEd), settlement (?v) and lateral wall displacement (Ux), which were calculated at the bending stiffnesses EI0, EI? and EIvar as a function of an axial compressive force N = 0 kN and N ? 0 kN for the following calculation models: - Walls only; hinged - Walls only; clamped - Walls and roof; clamped. EIvar based calculations turned out to be an iterative procedure. During this research two iteration procedures were developed to find the actual EI-distribution over the diaphragm wall height. An evaluation of the load distribution and cracked zones according to both iteration procedures, finally led to the conclusion that the results of iteration procedure 2 were valid for EIvar and that too for every calculation model. In iteration procedure 2 the actual EI-distribution over the wall height is obtained by considering the average M-line and average cracked zones based on EI0 and EI?. In this research a safety analysis was performed for: - The basic case (‘basic reinforcement ratio’ for the hinged and clamped case) and for; - The hinged case in particular, where a variation was made in the soil stiffness and the reinforcement ratio of the diaphragm wall. In the calculation models a ‘basic reinforcement ratio’ was the main input for determining EIvar. For the ‘Walls only; hinged’- model it was found that the packing structure was totally safe if the walls were designed based on EI?, while for the ‘Walls and roof; clamped’- model a safe structure was reached if the walls were designed based on EI0. The ‘Walls only; clamped’- model showed that a fully clamped connection is only an academic case, which is not realizable. The chosen connection type was found to have a major impact on the structural safety. Due to limited freedom of movement with regard to the reinforcement in the clamped case, variations were only studied for the hinged case. If stiffer diaphragm walls, achieved by applying a high reinforcement ratio, were used in the hinged case a model based on EI? still proved to be conservative. At a higher reinforcement ratio the behaviour tended even faster towards EI? (occurrence of greater cracked zones). When the soil properties (lower soil stiffness) were changed for the hinged case, EI? still proved to be a safe approach for the bending moment. However, for the deformations an EIvar based calculation was a safer method, because the actual deformations proved to be larger than what followed from the lower bound stiffness EI?. All analyses taken into account, it can be concluded that considering both the outer boundaries EI0 and EI? is not always a guarantee for a safe structure. Therefore, EIvar based calculations are necessary. It has been proven that the axial force (N) has no significant impact on EIvar. Especially, when designing the reinforcement a load distribution according to EIvar should be considered. Otherwise, there is a great risk of placing less reinforcement over a certain part of the wall, in particular for the hinged case. An EI0 based calculation for reinforcement design is only conservative if the maximum occurring bending moment of both walls is considered over the total wall height. However, this is a too safe approach. Of more practical relevance for (similar) future projects would be to design the reinforcement based on the ‘dekkingslijn’-principle using (1) the bending stiffness EI0, but then with 30-50% extra reinforcement at the bottom part of the wall or (2) the bending moment envelope based on different simulations with EIvar.