A four-point bending test was carried out by Kelleter (2022) on a simply supported concrete beam without actuators. The geometry of the beam and the test setup are shown in Figure 1A. A minimum bottom and top reinforcement avoid a premature cracking of the beam. The two-points vertical load is applied through a steel plate (IPE 100) on the top part of the concrete beam. The applied force is gradually increased from 0 kN to a maximum of 8 kN. The maximum displacement of the mid-node of the upper edge of the beam is measured at the end of the test. The horizontal strain profiles for the upper and bottom edges of the beam are also reported by Kelleter (2022).

A concrete beam with 10 integrated hydraulic fluidic actuators was tested by Kelleter (2022) under different loading conditions. The geometry of the beam is shown in Figure 1B. The test setup for the four-point bending test is the same as shown in Figure 1A. The steel actuators consists of two steel sheets and a spacer ring. The two outer sheets have a thickness of 1 mm, the inner ring is 2 mm thick. The inner sheet is ring-shaped and has a flange height of 5 mm, which prevents expansion in the radial direction (Figure 2A). Since the inner ring is significantly stiffer in the radial direction than the two outer sheets perpendicular to the plane of the sheets, hydrostatic internal pressure predominantly results in an expansion transverse to the actuator, i.e., along the longitudinal axis of the beam (see Kelleter, 2022). The function of the inner ring is to create a defined cavity. The three steel sheets are, after attaching a hydraulic pipe with an outer diameter of 4 mm from the same material, welded at the edge. Ignoring this pipe, the actuator is circular and has an outside diameter (D _A ) of 80 mm (see Figure 2A). The inner diameter is equal to 70 mm. Figure 2B shows the beam before casting reported by Kelleter (2022). The actuators are fixed in position via the pipes. In order to better secure the position, the beam is rotated by 180°. In the configuration shown, the pipes are routed laterally out of the cross-section.

The microplane material model with relaxed kinematic constraint (Ožbolt et al., 2001) implemented in the software MASA (Ožbolt, 1998) was used for the non-linear numerical simulations of the present study. In the microplane model, each FE integration point is ideally surrounded by a unit sphere discretized by a pre-defined number of microplanes with their normal directions (n) associated with microplane integration points on the sphere (Figure 4A). The material response is then computed based on the monitoring of stresses and strains in all the predefined directions, which are in this case 21 for the hemisphere, assuming central symmetry (Figure 4A). The microplane strains are assumed to be the projections of the macroscopic strain tensor ɛ _ij (kinematic constraint). According to a V-D-T split of the microplane strains (Figure 4B), the resulting strain vector on each microplane is decomposed into normal ( σ _N , ɛ _N ) and two shear components ( σ _M , σ _K , ɛ _M , ɛ _K ). The normal microplane stress and strain components are decomposed into volumetric and deviatoric parts ( σ _N = σ _V + σ _D ; ɛ _N = ɛ _V + ɛ _D ). As discussed in Ožbolt et al. (2001), the microplane model with relaxed kinematic constraint accounts for the strong localization of strain in case of dominant tensile loads. From the physical point of view, when the strain localizes in one direction (direction of crack opening), the stress oriented in the same direction decreases (material softening). Simultaneously, an elastic decrease (unloading) is observed for the strain and stress components oriented predominantly laterally to the direction of crack opening. To account for this mechanism a discontinuity function (ψ) is applied to the deviatoric and shear microplane strain components to reduce all stress components to zero after cracking. The function is controlled by the maximum principal strain and volumetric stress-strain relationship.

The stress-strain relationships at the microplane level are based on the uniaxial damage theory, suitable for the non-linear analysis of heterogeneous materials, such as concrete and wood (Gambarelli and Ožbolt, 2021; Ožbolt et al., 2022). The initial elastic properties of the material are controlled by the initial elastic modulus (E) and Poisson’s ratio (ν). The exponential damage functions defined for the V-D-T microplane stress-strain relations (Ožbolt et al., 2001) depend on a set of microplane parameters, namely: a, b, p, q for the volumetric compression; e1, m for the volumetric and deviatoric tension; e2, n for the deviatoric compression; e4, k for the two shear components. The adopted elastic (E = 48,000 MPa, ν = 0.2) and microplane parameters (a = 0.003, b = 0.05, p = 0.8, q = 1.5; e1 = 0.00004, e2 = 0.0015, e3 = 0.0011, e4 = 5.0, n = 0.57, n = 0.97, k = 0.65), have been properly calibrated on the single unit size finite element (microplane model) to correctly reproduce the macroscopic elastic and fracture behavior of the C35/45 concrete used in the experiments for the reference beam, as well as for the beam with actuators (configurations M1, R2, U1). The resulting uniaxial tensile and compressive curves are respectively shown in Figures 4C, D. The non-linear material model for concrete was first validated against the reference beam and then used in the beam with actuators for the three configurations reported by Kelleter (2022).

The macroscopic material properties used in the numerical simulations of the present study are summarized in Table 3. It is worth mentioning that a linear elastic behavior was assumed for the actuators (steel S235), while a bilinear elasto-perfectly plastic constitutive law was employed for the reinforcement (B500B) (see Kelleter, 2022).

As mentioned above, only one-quarter of the beam was modeled (see Figure 3), therefore symmetry boundary conditions were imposed in the x-y and y-z plane. The mid-line of the bottom steel plate was fixed in the vertical (y) direction (simply supported beam). The load was applied on the top of the steel plate. The pressure in each actuator was applied on the inner surface of the actuator along the longitudinal axis of the beam. The values of the pressure in each actuator vary according to the configurations discussed in the previous sections.

The geometry of the actuators and the surface area on which the pressure was applied, have a significant impact on the deformational behavior of the beam. As explained by Kelleter (2022), for a realistic evaluation of the pressure applied in the actuators, a pressure leak between the area of the actuator ring and the outer part should be considered. This is related to the fact that the steel sheets of the actuator are welded from the outer edge and possibly the hydraulic oil has leaked into the inner area between the stiffening ring and the outer plate. Therefore, more surface area is available for the pressure application. Figure 5A shows the FE discretization of the actuator and the corresponding deformation is shown in Figure 5B. A 2 mm pressure leak was assumed between inner and outer surface of the actuator, i.e., the effective area for the pressure application corresponds to D _in = 78 mm, instead of 70 mm.

The microplane material model is able to correctly reproduce the deformational behavior experimentally obtained for the reference beam. Figure 6A shows the strain distribution along the longitudinal axis of the beam (z direction) with positive strains in the lower part of the beam and negative strains in the upper part, indicating the tensile and compressive zones in the concrete respectively.

The microplane material model with calibrated parameters is able to predict near displacement results. This can be observed in Figure 7 from the load–displacement curve for both the numerical and experimental results, where the displacement of the mid-node of the upper edge of the beam is monitored. The displacement predicted by the microplane material model is −0.0604 mm and the experimental displacement is −0.062 mm. Therefore, the error in the predicted value at the end of the simulation is found to be 2.6%. In the numerical simulation performed by Kelleter (2022), a mid-node displacement of −0.081 mm is obtained, with 23.5% deviation from the experimental result.

The maximum strains of the mid-node for the upper (−0.0047%) and lower edge (0.0062%) of the concrete beam are also compared with the experimental results (−0.0049% and 0.0061%). The errors in the strain values are 4.3% for the upper edge and 1.6% for the lower edge. In the numerical simulation by Kelleter (2022) respective strains of −0.006% and 0.0062% are determined, with corresponding errors of 18.3% and 1.6%.

It has to be mentioned, that Kelleter (2022) used the FE model with actuators to calibrate the material model and to simulate the reference beam case. Also, Kelleter (2022) used the isotropic linear elastic constitutive law for concrete and steel in the reference beam case.

As a part of the non-linear analysis, the four-point bending simulation in MASA was continued beyond the 8 kN to observe the cracking behavior of the beam. The resulting damage (cracking) distribution is shown in Figure 6B in terms of maximum principal strains, where the red zone in the legend corresponds to a crack width of 0.1 mm. Similar as in the experiments, the cracks originate in the tension zone of the concrete at a load higher than 35 kN.

As reported in Section 2.2.1, in the R2 experimental test a pressure of 5 bar is first applied to the actuators to compensate the self-weight of the concrete beam. A consequent positive displacement of the mid-node in the upper edge of the beam of 0.014 mm is measured in the experiment. Thereafter, the vertical load and the pressure in the actuators are applied simultaneously, according to Section 2.2.1. It is worth mentioning that high scatter in mid-node displacement was obtained in the experiment by Kelleter (2022); therefore, a value of −0.001 mm was here calculated from the experimental displacement profile as average of different points at the end of the test (full load and pressure applied, i.e., adaptive state).

The same as in the experiment, an initial pressure was also applied in the here presented numerical model and the obtained distribution of vertical displacement is shown in Figure 8A. The displacement of the mid-node in the active state in the simulation with the microplane material model is equal to 0.0157 mm. Thereafter, the vertical load and the pressure in the actuators were applied simultaneously in steps of 10 bar, until reaching the maximum load and pressure values specified in section 2.2.1. The obtained results for the adaptive state are shown in Figures 8B, C. It can be seen that the mid-node displacement at the end of the numerical simulation (adaptive state) is equal to −0.025 mm, with deviation of 96.0% with respect to the experiment. In the numerical simulation performed by Kelleter (2022), a mid-node displacement of −0.024 mm is obtained for the adaptive state, with an error of 95.8%.

The different lateral expansion of the actuators induced by the different applied pressures is also shown in Figure 8C. The maximum vertical strains of the mid-node for the upper (−0.00206%) and lower edge (0.00649%) of the concrete beam are also compared with the experimental results (−0.00204% and 0.0070%). The error in the strain value is 0.8% for the upper edge and 7.9% for the lower edge. Kelleter (2022) obtained respective strains of −0.002% and 0.0046% in the numerical simulation, with corresponding errors of 2.0% and 52.2% in relation to the experiment.

It has to be pointed out that a more realistic modeling of the steel actuators was adopted in this study by considering also the oil leakage into the inner area between the stiffening ring and the outer plate.

The aim of this test was to check the failure of the beam. Therefore, the vertical load and the pressure were significantly increased. It is worth mentioning that an already tested beam was used in the experiments. Therefore, the presence of internal pre-damage cannot be ruled out.

In the numerical simulation with the microplane material model, the first crack was detected at a load of 14.4 kN, whereas in the experiment the first crack was observed at a load of 12.85 kN. The small deviation can be due to the pre-tested beam used in the experiments. The cracks distribution at the end of the numerical simulation with the microplane material model is shown in Figure 9B in terms of maximum principal strains, where the red zone corresponds to a crack width of 0.1 mm. It can be seen that the beam cracked over the entire height in the area of the inner actuators with good agreement with the experimental results (Figure 9A). The cracks localization in the center of the beam is related to the higher pressure of the inner actuators.

The non-linear numerical simulation was also performed by Kelleter (2022), by using the elasto-plastic damage model in ABAQUS. The cracking pattern obtained with the elasto-plastic damage model is also in agreement with the experimental results. However, no information is provided regarding the load and pressure values at which the first crack forms.

As mentioned in the previous sections, in this test equal pressure is gradually applied to each actuator to compensate the passive deflection of the beam. In the experiment, a near zero displacement of the mid-node in the upper edge of the beam (adaptive state) is detected for a pressure value slightly lower than 50 bar. The pressure is then further increased to check the failure of the beam. The first cracks in the outer actuators are detected at a pressure value of 60 bar. Smaller cracks are also visible at the neighboring actuators. The test is stopped at a pressure value of 97 bar.

The numerical simulation with the microplane material model gave a near-zero displacement for the mid-node in the adaptive state for a pressure value in the actuators slightly higher than 80 bar. The vertical displacement distribution in the passive (mid-node displacement of −0.0624 mm) and adaptive state (mid-node displacement of 0.00078 mm) are shown in Figures 10A, B, respectively.

A comparison between experimental and numerical results in terms of variation of the mid-node deflection with the applied pressure is shown in Figure10C. The significant deviation between the curves can be attributed to different aspects. It is worth mentioning that only one test was performed experimentally, which renders a clear interpretation of the observed trend difficult. One aspect could be pre-damage of the beam due to several reasons, but additional experimental tests are needed to fully clarify the causes for these deviations. Also, in the numerical FE model, the pressure is applied on a surface area of 4775.94 mm² (diameter 78 mm), by considering 2 mm for the thickness of the actuator. Since the steel sheets are welded from the outer edge, it is difficult to precisely estimate the effective area for the pressure. If less area is considered in the model, higher pressure is required in the numerical analysis to fully compensate the displacement of the mid-node.

Another important aspect for the adaptive behavior of the beam is the stiffness of the actuators. To better understand the influence of the area for the pressure application and the role of the actuators’ stiffness, the actuators were deleted in the model by leaving the corresponding holes within the concrete. As a consequence, the pressure was directly applied to the concrete surfaces with area 5,024 mm ² (diameter 80 mm), corresponding to the external surfaces of the actuators (see Figures 11A, B).

The new obtained results are shown in Figure 11C against the experimental ones. A better agreement can be observed, confirming the importance of the surface area for the pressure application. Similar to the experiments, a zero mid-node displacement is observed for a pressure value of 50 bar. This result is also consistent with the numerical one obtained by Kelleter et al. (2020) for the same configuration without explicit modeling of the actuators.

In the experiment, the pressure in the actuators was further increased to observe the cracking in the beam (Figure 12A). The obtained numerical crack pattern (model with actuators) is shown in Figure 12B for the front and in Figure 12C for the internal sections, showing a good agreement with the experimental results. It can be seen that the equal pressure in the actuators induces a first cracking in the area of the outer actuators. At the end of the test, the crack goes through the complete height of the beam.

The cracking pattern obtained by Kelleter (2022) for this configuration is also in agreement with the experiment; however, no information is given regarding the load and pressure values at which the first crack forms.