Numerical models

Another method to calculate the load-rotation curve is the numerical approach. Either a numerical method can be applied to simply solve numerically the differential equation of an axisymmetric slab considering the nonlinear slab behavior or a more sophisticated numerical method can be used such as the finite element method to model the slab as a whole. Guandalini (Guandalini 2005) uses a numerical approach in order to numerically solve the axisymmetric case. For this, he calculates the forces acting on a small slab element as a function of the state of strain in the slab. By using the equilibrium conditions within a discrete element (Figure 2.19) and by applying the boundary conditions, he obtains the response of the axisymmetric slab.

Figure 2.19: Solving of axisymmetric cases by numerical integration

However, for more general geometries and loadings, an axisymmetric solution might not be suitable. In these cases an approach based on the finite element method is more promising. Starting in the last century, various research has been performed regarding the application of the finite element method for reinforced concrete. Due to the enormous amount of different developments in this area, it is neither possible nor the objective of this research to present the development of elements used for the modeling of reinforced concrete in its entirety. Therefore, the following sections present a selection of finite elements analyses and developments.

Δφ mtiּΔr mtiּΔr mriּriּΔφ viּriּΔφ (mri+Δm)ּ(ri+Δr)ּΔφ Δr ri (vi+Δv)ּ(ri+Δr)ּΔφ qּAi

Generally, the elements chosen for the analysis of flat slabs are either plate (or shell) elements or 3D solid elements. Whereby the latter is only used for the modeling of a limited portion of the slab (e.g. punching test specimens) since the modeling of a flat slab requires a large amount of elements leading to an extensive need of computational resources and calculation time. The advantage of the use of 3D solid elements is that the concrete and the reinforcement can be modeled separately so that the modal accounts of the orthogonal reinforcement. Additionally, the punching shear reinforcement can be implemented in the same model allowing the analysis of the load transfer path in the column vicinity. With respect to punching of flat slabs, Beutel (Beutel 2003) and Häusler (Häusler 2009) investigated several punching tests with and without shear reinforcement by using models with 3D solid elements. The analysis led to acceptable results regarding the response of the investigated test specimens with and without shear reinforcement. Nevertheless, it has to be mentioned that a calculation using 3D elements is sophisticated so that the input and model parameters have to be chosen carefully and the details such as the anchorage of the shear reinforcement (Beutel 2003) have to be appropriately modeled.

A simpler approach is the use of 2D elements such as plate or shell elements. However, the main challenge by using plate or shell elements is the implementation of the nonlinear response of a reinforced concrete section. Several methods exist that can account for the nonlinearity in the 2D finite element analysis. One possibility is the use of elements that consist of different layers which are separately integrated (Figure 2.20) and later assembled for the whole element. Thus, every layer has its own stiffness matrix which depends on the state of deformation. Formulations for this method with respect to plate and shell elements have been developed, amongst others, by Hand et al. (Hand et al. 1973) and by Vecchio and Polak (Vecchio 1989; Polak 1992; Polak and Vecchio 1993). The main difficulty lies in the definition of the torsional stiffness. For this, Hand et al. (Hand et al. 1973) introduced a shear retention factor that accounts for dowel action and aggregate interlock so that the element can provide shear stiffness. Polak (Polak 1992) defined the torsional stiffness as a function of the concrete stiffness in the principal directions.

Figure 2.20: Layered element

Steel stresses

Concrete stresses

Another possibility to introduce the material nonlinearity into the finite element method is by using a modified stiffness approach. This method was introduced by Jofriet and McNeice (Jofriet and McNeice 1971) and by Bell and Elms (Bell and Elms 1972). In this method, the constitutive relationship of a reinforced concrete element is calculated in advance leading to the secant stiffness of the element depending on the state of deformations. Afterwards, the secant stiffness is used for a linear finite element calculation leading to a new state of deformation for each element. This routine will be repeated until a certain tolerance is met. The main advantages of this method are firstly the calculation speed since the element response has to be calculated only once and secondly the robustness of the calculation. Due to the fact that the constitutive relationship is calculated in advance, the response of the element is well defined so that for each state of deformation, a defined stiffness exists.

Vaz-Rodrigues (Vaz Rodrigues 2007) and Tassinari (Tassinari 2011) used this method with respect to shear test and punching test calculations, respectively. The main problem however was the introduction of the torsional stiffness after cracking. Similar to the shear retention factor introduced by Hand et al. (Hand et al. 1973), Vaz-Rodriques (Vaz Rodrigues 2007) introduced the torsion retention factor to account for the torsional stiffness of the slab (Figure 2.21). For regular punching tests Vaz Rodrigues (Vaz Rodrigues 2007) and Tassinari (Tassinari 2011) empirically determined the torsion retention factor to be βt = 1/8. Since this value is completely

empirical and thus it is only valid for the geometries and loadings according to the tests on which it is based on, a more sophisticated formulation is desired. Chapter 5 presents a constitutive model that allows modeling the bending and the torsional response of a reinforced slab element.

0 0 0 0 0 0 ∙ 1 ∙ ∙

∙

Figure 2.21: Definition of the stiffness matrix using the torsion retention factor βt (Tassinari 2011) m mR EI1 1 EI0 1 χ Dx,i or Dy,i 1 χi mi mcr