The spatial finite element model idealises three-point-bending tests of RC beams under biaxia loadings (Figure 2). Its boundary conditions and force applications are predefined and the symmetries of geometry and load are utilised to halve the model structure and thus save computing times. Figure 1. Notation, generalised treatment of biaxial shear forces. Table 1. Model parameter. Vertical forces V,,., = V, a referred to by ay = 0, while a diagonally directed shear force V,,, yields ay = 1. It is indispensable to hold 0 < ay < 1, otherwise the notations of V,, V., h and b have to be exchanged. Figure 2. RC structure and FE model with parameters. "Compression only” springs assure a numerically stable changeover of forces from the solid beam to the steel supports. They also exclude unrealistic tensile bearing reactions. The concrete body is discretised in , x n, elements in the section plane and 1; +8 (4) elements in the longitudinal direction. They represent the "host elements” in the applied concept of “embedded elements”. 8- node C3D8 or 20-node C3D20 solid elements with linear or quadratic interpolation functions are chosen from the ABAQUS element library. T3D2 truss elements — lying embedded in the concrete Figure 3. Stress-strain relation for (cyclic) compressive loading, experiments acc. (Sinha et al., 1964). The first two sections describe the ascending branch up to the peak load f,,, at &1. Their formulations are similar to the recommendations of the Model Code (CEB-FIB, 1993). The third and descending branch takes account for its dependency on the specimen geometry (Vonk, 1993; Van Mier, 1984) to ensure almost mesh independent simulation results. Thus, o.(3) incorporates within the descent function y, the constant crushing energy G,, (Kratzig & Pélling, 2004) as a material property in addition to an internal length parameter /. derived from the grid structure of the element mesh. The evolution of the compressive damage component d, is linked to the corresponding plastic strain ¢?’ which is determined proportional to the inelastic strain &" = & - oF. using a constant factor b, withO <b. < 1. Figure 4. Stress-crack opening and stress-strain relations for (cyclic) tensile loading, experiments acc. (Reinhardt & Cornelissen, 1984). using the principle of the "Fictitious Crack Model" (Hillerborg, 1983). Thus, a product of the inelastic strain and an internal length parameter /, replaces the crack Openine) w to yield o, = of(w= Lei" =1(& - oE.')) and w is smeared over the average element length /,= V.”. As intended, o( ¢) then encloses the ratio of fracture energy G; and /, (Bazant & Oh,1983). Similar to (6) the damage d, depends on ¢?” and an experimentally determined parameter b, = 0,1 (Figure 4, right). So, unloading is assumed to return almost back to the origin and to leave only a small residual strain. Figure 6. Comparisons of bi- and triaxial strength results to experimental data. Table 2. Material parameters, * uniaxial loading, ** multiaxial loading. The following parameters are adopted for the reinforcing steel of stirrups and longitudinal bars: E, = 200.000, £,; = 1111, f, = 500 (all values given in [MPa]). Figure 7. Load- displacement response and specimen details, experimental data acc. (Mark, 2004). Figure 8. Load-displacement response and specimen details, experimental data acc. (Toongoenthong & Maekawa, 2005). Figure 8 shows similar simulation results for the recalculation of the test data of (Toongoenthong & Maekawa, 2005), where /.= 2000 mm. Peak load and even the descending branch acceptably agree with the experimental ones, if the element mesh does not get too course. The overestimation of initial stiffness properties remains. an indirect way. Hence, experimentally derived crack pattern at the lateral surfaces of the concrete body are displayed together with the distributions of concrete and stirrup stresses, both calculated for the peak load. Compressive concrete struts nucleate arch-like or strait inclined with an average angle of about 40° to the girder axis (white areas of the stress distributions, broken lines in the truss model), as concrete cracks in tension. They lie almost parallel to the surface cracks. Stirrups take over vertical tensile stresses and pronounced deflections as well as redistributions of inner forces take place. As expected, stirrups yield localised, just where cracks cross. Figure 9. Experimental crack pattern and calc. distribution of concrete stresses (top), stirrup stresses and experimental crack pattern, truss mechanism (bottom). Figure 10. Comparisons of biaxial shear resistances to numerical and experimental data. The shear resistances V,;,, are extracted from the maximum forces Fig, = 2V sin Of the simulations and compared to the ultimate shear resistance Vp; (design formula) of the tensile strut (Figure 10). Experimental ratios V..,/V3 of several uniaxial and two biaxial shear tests (Mark, 2004) are added. Effects of the load inclination are removed by the mechanical reinforcement ratio @,, that adjusts the ratios Vein/Vp3 and V..,/Vp3 on the level of uniaxial shear and thus allows comparisons of uniaxial and biaxial results. Figure 11. Effect of additional stirrups in cases of distributed longitudinal bars. Biaxial shear exhibits its own specific characteristics. Typical features are the formation of complex, three dimensional distributions of concrete stresses with stiffening effects in the stirrup’s corners or a more localised and side concentrated yielding of stirrup legs. An additional one is illustrated in Figure 11: As compressive shear struts rest on the longitudinal bars under tension, they bend aside those bars and thus the stirrup legs, if the bars are located at the section sides and not at the corners of the stirrups. Additional, horizontally oriented stirrups help to improve the situation. They reduce lateral deformations and increase bearing capacities.
![Table 2. Material parameters, * uniaxial loading, ** multiaxial loading. The following parameters are adopted for the reinforcing steel of stirrups and longitudinal bars: E, = 200.000, £,; = 1111, f, = 500 (all values given in [MPa]).](https://figures.academia-assets.com/36867529/table_001.jpg)
