Collapse analysis of externally prestressed structures

Text Sample: Chapter 3.4.6, Ordinary reinforcement: The ordinary reinforcement is provided by the means of rebar elements. These elements can be defined singly or can be embedded in oriented surfaces within an at least two-dimensional element. The latter method is used here. Section 3.4.4 explains the configuration of the element type. This section deals with the amount of reinforcement needed and how it is applied. The ordinary reinforcement is of high importance for the model. There are two reasons. The first one is the nature of the structure to simulate. It is a reinforced concrete structure. This might sound trivial, but it is often tried to model the concrete as a homogeneous isotropic material. As long as there is no serious" crack in it, there is every justification to do so. However, reinforced concrete and prestressed concrete overcoming the tensile strength of the concrete must be modelled with any reinforcement. Serious" cracks have therefore two characteristics. They have to be open, and they do not transfer any stress component from the principal strain normal to the crack propagation. Both of these conditions are true in certain areas of this model. The other reason, why reinforcement is needed, is the constitutive model for concrete used in this analysis. This model does not or hardly converge without any reinforcement. The manuals of ABAQUS mention this phenomenon. Smeared reinforced concrete models own this characteristic. This was also found during the analysis. The amount of ordinary reinforcement is taken as 1.5%. The magnitude is calculated from EC2 (DD ENV 1992-2:2001). The calculation can be followed in Appendix B: Calculation of the minimum reinforcement. The same amount is taken for the flanges as well as for the webs. The reinforcement mesh is orthogonal placed in the midsurface of the concrete shell. For reasons of simplicity, no attempt is made to place the reinforcement at its right position closer to the surfaces of the shells. Although the reinforcement input has the information of the diameter and the spacing, the actual calculation is performed only with a smeared orthotropic layer. Hence, the value 1.5% reinforcement and the orthogonal directions are sufficient to describe the actual input. But this amount is important because it describes a major difference between the bridge types. The monolithic types have this reinforcement throughout the whole structure. The precast segmental type has the continuous reinforcement only in the segments. It stops at the joints. This is an important fact, because this investigation aims to find out, if this has any significance. Chapter 3.4.7, Prestress: The section deals with the application of the prestress in the FE models. There are two methods of doing it. The prestress can be applied as external forces on the model and the constitutive model of the prestressing steel has to be adjusted by lowering the yield point of the prestressing steel. This is the recommended method from several authors, for instances Kotsovos and Pavlovic. The other method is to apply a real prestress to the tendons with initial conditions. The second method is chosen in this analysis. The first method represents the classical method of doing it and involves deviations from the original model, which are not necessary with FE technique available nowadays. The main shortfall is the unsatisfactory modelling of the behaviour of the model at the anchorages. The anchorage load will be typically underestimated. This might be satisfactory in a high number of cases, because it can be considered as a local effect only. The second approach, applying an initial prestress to the tendon, is the more rigorous analysis. One of the aims of this analysis is to find out the tendon stress increase at ultimate state. This can be easily done with this method. The tendon stress can be read at failure and compared to the initial stress. Besides, this is exactly, what is found in reality. The method with the substitute forces allows not such a direct comparison. However, the direct method has also several difficulties. The first is applying the right prestress. The problem is the elastic shortening of the structure, which means if the prestress is applied to the tendon and via the anchorages to the structure, the concrete bridges shorten elastically. The effect can be fond in reality in pretensioned structures only. This does not happen with post-tensioned structures. Typically the jack is used to apply the prestress wanted. In fact, the elastic shortening is even not noticed during the jacking process, because the jacking force is applied directly against the structure. Elastic shortening is also depended from the dead load and the position of the anchorage itself. ABAQUS offers here an easy way of keeping the prestress constant during the initial equilibrium stage. A function is available, which is called *Prestress hold". This method works excellent with the internal prestressed version and the externally prestressed version with blocked deviators. The prestress can be set to the wished value and is kept constant during the initial equilibrium state. The initial equilibrium state consists mainly of applying the prestress, the dead load and occurrence of any kind of shortening. However, this does not work for the pure external type and the precast segmental version. The allowance of movement at the deviators makes the difference. If an initial stress is applied to the tendon and the first equilibrium step is started, the tendon moves at the deviator as long as it reaches equilibrium for any reason. This lead in trails to tendon movements of several meters by applying the prestress and the dead load only (Figure 3-36). Of course, this does not happen in reality. The solution for this problem is a more empirical search for the initial prestress. The tendon is stressed considering elastic shortening, which means higher than original wished. The function *Prestress hold" is omitted, elastic shortening occurs and the prestress reduces. The Starting prestress can be found with some trials or by simple hand calculations considering elastic shortening. It might be worth mentioning that the distribution of tendon stress in reality even at initial state is varying along the length mainly due to friction and deflecting forces, and is still more complicated than simulated in this model. But it is believed that the calculation undertaken are a reasonable approximation. The second problem initiated by this calculation approach is the concentration of stresses at the anchorages. This might sound logical and happens in reality. Hence, it is also an advantage, because it is close to the real condition. But the high stress at the anchorages caused a series of anchorage failures during the analysis. The anchorage zone had to be heavily reinforced. The amount of reinforcement is not further specified. The amount at the anchorage zones was simply increased until the anchorage were save, because it is not the aim of this calculation to investigate the anchorage zones. The elements for the tendons are beam elements as specified in 3.4.4 with a very low second moment of area. The tendon element itself has a Young's modulus of 0.1 N/mm². It is effectively a rubber tendon. But this rubber tendon is reinforced with a rebar element, which has the properties of the real tendons. The combination is caused from the restrictions imposed of ABAQUS. It was not possible to prestress any element. But it was possible to prestress a rebar element, but a rebar element needs a parent element. The area for each long tendon is 28500mm²/2=14250mm² and for each short tendon is 3600mm²/2=1800mm². The Young's modulus is 193000 N/mm². These properties are further explained in section 3.3. The applied prestress for the internal type and the external type with blocked deviators is 1197.6 N/mm². The applied prestress for the other two versions is 1275 N/mm². The effect is the same and reasons explained above. Other information about the time dependent application of dead load and prestress is available in the next point. Chapter 3.4.8, Material and geometric non-linearity: All calculations have been carried out firstly linear. This tends to be of significant importance. Firstly, the mesh can be verified for sensible stress magnitude output and sensible boundaries of the stress isobars. Critical points for the later non-linear analysis can also be seen in this pre-analysis. The source of non-linearity in this case is mainly material non-linearity. However, considerable deformation can also be expected. This makes the use of a combined non-linear calculation algorithm necessary. A non-linear calculation is characterised by serious of linear steps divided into increments and iterative steps. The prerequisites for the calculation of material non-linear analysis are prepared by introducing non-linear constitutive models (3.4.5). This analysis is divided into two steps. The dead load and prestress are applied in step 1. The second step is the application of two point loads over the webs and the middle deviators until the structure fails. The first step uses the method of Newton and the second step uses the method of Riks. The reason for changing the methods is the higher capability of the Riks algorithm. This method is even able to cope with sign change of the stiffness. As already mentioned, the first analysis step involves applying of the prestress and the self-weight. This has to be done careful because an unbalanced apply of those components could already destroy the bridge or damage the elasticity of the model with unreasonable high cracking at an early stage. Cracks are stored for the whole calculation, which means in practice they do not vanish during the life of structure but can only growth. Further explanation on damaged elasticity can be found in section 3.4.5.2. The introduction of the prestress does not typically damage the elasticity of structure in reality. The economical Newton's method can be used here. This algorithm converges quite rapidly but is not capable of dealing with qualitative stiffness change. Such a process is not expected by application of the prestress, because the concrete tensile strength will not overcome anywhere in the structure and no cracks influencing the calculations should growth. The original method of Newton is slightly modified in this calculation. The calculation is divided into increments. The next figure demonstrates the convergence scheme used. Only a short introduction to the method will be given. Further formulations are omitted here, because they are readily available in Finite Element textbooks, for instance Zienkiewics and Taylor.