Linear elastic model for hybrid beams

The first finite element model for FRP-concrete hybrid beams was conceived as an easy-to- implement numerical solution where the structural components are reduced to their geometrical essence and where the constitutive materials and connectors behave in a linear elastic way. Consequently, the partial interaction at the interface between the concrete slab and composite profile was based on the elastic stiffness of the connectors as determined from push-out tests or empirical formulations (Eq. (4.18)). This approach is different to the analytical model described in the previous chapter, in the sense that the connection is represented by a series of discreet finite elements and not by a continuous interface with uniform stiffness. Therefore, it allows for a more accurate depiction of structural members with mechanical shear connectors and with varying grades of connector spacing.

On these considerations, a three-dimensional finite element model was developed in the structural analysis software SAP2000 [209] that served as basis for the M2 hybrid beams tested by the author. Similar models were also created for the referenced beams used in the validation of the analytical equations (section 4.7). The concrete slab was modeled as a thick shell which includes transverse shear deformations in the out-of-plane plate-bending behavior, following the thick-plate Mindlin-Reissner formulation. Longitudinal slab reinforcement bars were modeled as smeared layers in the definition of the shell elements; however, the results showed that their inclusion has a minimal influence over the global flexural response of the hybrid beams. The composite profiles were idealized as frame elements based on Timoshenko’s beam theory while the shear connectors were modeled as a single condensed row of link elements positioned at the specified physical coordinates, as seen in Figure 5.6.

Figure 5.6: Schematic view of the deformed hybrid beam M2 finite element base model.

Since material nonlinearity is accounted by the software only at the location of concentrated plastic hinges and when using layered shell objects, the structural components were assigned elastic material properties according to the available data. Concrete was considered to be homogeneous, isotropic and was defined by its elastic modulus and Poisson’s ratio. The composite profiles were designed as homogeneous with orthotropic elasticity and with defined material orientation. In the case of the M2 hybrid beams, the GFRP input values were the same as in the profile calibration tests previously discussed. Shear connectors were characterized just by their elastic stiffness and were allowed to deform only in the principal direction of the beam. For complete shear interaction modeling, the movement of the links was restricted in all directions.

The member’s top shell was meshed with quadrilateral elements with a maximum size of 50 mm, and further divided at the load application regions. The bottom frame was meshed automatically at the joints formed by the intersection with the connector links. One of the supporting ends of the beam had all three translations fixed while the other support had all but the axial direction restrained. Several trials of the load application method, either on small areas, element edges or central nodal points, proved that there is little difference amongst them, so the latter method was adopted. As hybrid beams model M2 shared the same geometry, the loading cases and elastic properties were necessary to be changed in-between simulations.

A static linear analysis was performed for each one of the considered specimens using the maximum recorded force from the experiments. Henceforth, geometrical nonlinearities from second order effects such as large displacements and large deformations were not included for these initial models. In addition, to simplify the simulation procedure, the analysis was limited to the main longitudinal plane of bending by restricting the other degrees of freedom available.

Under these conditions, the midspan deflection results of the referenced hybrid beams are presented in Table 5.1 and discussed. The experimental and analytical results at failure act as a validation tool for the FE numerical model. For comparison reasons, both complete and partial shear interaction states were investigated, and the concrete slab was assumed to be uncracked in this first analytical model. Exceptionally, for this calculation, the exact 𝜉𝑖,𝑚𝑖𝑑 parameter was used instead of the simplified 𝜉𝐸𝐶5.

Table 5.1: Comparison between experimental, analytical and finite element analysis midspan deflections (𝒘𝒕) computed at failure considering a state of complete (𝒘𝒕𝒄𝒐) or partial (𝒘𝒕

𝒑𝒂

) shear interaction.

Beam Experimental Analytical (uncracked slab section) FE analysis 𝑤𝑡 (mm) 𝑤𝑏𝑐𝑜 (mm) 𝑤𝑠ℎ (mm) 𝜉𝑖,𝑚𝑖𝑑 𝑤𝑡𝑐𝑜 (mm) 𝑤_𝑡𝑝𝑎 (mm) diff. a (%) 𝑤𝑡𝑐𝑜 (mm) diff. b (%) 𝑤_𝑡𝑝𝑎 (mm) diff. b (%) diff. a (%) HB1 92.8 49.1 9.1 0.153 58.2 65.7 -29 58.0 -0.2 64.6 -1.6 -30 HB3 21.0 6.2 6.7 0.636 12.8 16.8 -20 12.9 +0.6 16.4 -2.3 -22 B7 70.4 45.3 4.6 0.282 49.8 62.6 -11 49.7 -0.2 61.2 -2.2 -13 SP2 33.4 15.2 4.8 0.185 20.1 22.9 -32 20.0 -0.4 22.4 -2.0 -33 No1 12.6 7.2 4.9 0.412 12.0 15.0 +19 12.0 ±0.0 14.7 -2.1 +16 No2 16.6 9.0 5.3 0.341 14.4 17.4 +5 14.3 -0.5 17.1 -2.1 +3 a _{Difference versus experimental value.}

b _{Difference versus analytical value.}

During the post-processing of the results, it was observed that the simple computational model failed to capture appropriately the shear deformations of the composite members, although individually (i.e., per component) it evaluated them correctly. In consequence, since the partial interaction effects impact mostly the bending deflection contribution 𝑤𝑏 and not the shear deflection component 𝑤𝑠ℎ, the analytical value of the latter was added to the bending deflection computed in the FE analysis. The differences between the numerical and analytical predicted deflections prove to be very small – around 2% for the partial interaction scenario and nearly undiscernible for the complete shear interaction case.

Nevertheless, the differences against the experimental values are significant for both theoretical models due to the absence of nonlinear features. Thus, the flexural response of the FE model is substantially more rigid, as seen from the load-deflection charts plotted in Figure 5.7 for the referenced hybrid beams. This time around, the analytical model included for comparison assesses the flexural stiffness of the beams based on the reduced concrete section due to cracking and on the simplified dimensionless parameter 𝜉𝐸𝐶5. As seen, the linearly-elastic finite element model may be suited only for serviceability analyses or for simple checks at initial load values, when the concrete slab is not severely damaged.

Figure 5.7: Numerically predicted flexural responses of referenced hybrid beams versus experimental curves and analytical

In a second comparison, the load-deflection curves of the M2 hybrid beam models tested by the author are plotted in Figure 5.8. Once more, the FE model is able to follow the experimental response only at an initial stage. After the concrete slab starts cracking and the bolts begin yielding, the linearly- elastic model diverges abruptly from the test. Initial interaction effects caused by adhesion, friction or pretension of the bolts were neglected throughout the simulations.

Figure 5.8: Numerically predicted flexural responses of hybrid beams M2 versus experimental curves and analytical

estimations. Concrete cracking and partial interaction were accounted for in the analytical model.

A last analogy was made between the numerical, analytical and experimental slip distributions along the interface of the M2 hybrid beams, at an intermediate load value of 50 kN. The charts illustrated in Figure 5.9 demonstrate a good equivalence for the two theoretical models and furthermore indicate that the relative end slip measured during the tests was close to the predicted results. Notwithstanding, there are slight differences toward the supports, where the FE model displays lower slip values than the analytical calculations. Finally, it was observed that regardless of the static scheme applied, three-point bending or four-point bending, the analytical and numerical distributions had a similar degree of correlation.

Figure 5.9: Numerically computed longitudinal slip distributions in hybrid beams M2, at an intermediate load level of 50 kN,

versus experimental data points and analytical predictions.