Finite Element Modelling of Flat Slabs Strengthened with Shear Bolts with Small and Large Anchorages

Slab-column junctions in flat slabs without shear reinforcement are highly susceptible to punching shear failure, a kind of brittle failure that occurs without prior warning signs. In certain cases, the punching shear strength of these junctions may be insufficient due to many circumstances, including changes in building usage, design or construction mistakes, reinforcing corrosion, and concrete degradation. Typically, this failure occurs when the externally applied shear stresses surpass the tensile strength of the concrete. Therefore, it is essential to recommend appropriate shear reinforcement for this purpose. Numerous researchers over years have proposed and investigated various punching shear reinforcements for new and existing flat slabs.

Numerous researchers performed experiments in testing and evaluating different kinds of shear reinforcements for the new flat slabs, such as stirrups, bent-up bars, shear studs, truss-shear reinforcement, and shear heads. Shear reinforcement can be added around the column where bending and shear stresses are at their highest levels to enhance the shear resistance and ductility of the slab. Shear reinforcement is positioned to intercept the inclined cracks within the slab depth. Bent-up bars were utilised by references [1, 2]. Closed stirrups were examined by references [3-7]. Shear heads were examined by references [8-10]. Shear studs were utilised by references [11, 12]. Pre-assembled truss bars were deeply investigated by reference [13].

There are three distinct approaches to strengthening existing flat slabs: 1) By expanding the supporting territory, hence increasing the critical section around the column [14-16]. This would improve the flexural capacity of the slab. Nonetheless, the slab-column connections demonstrate comparatively more brittle behaviour. 2) Implementing post-installed shear reinforcements [17-21], including shear bolts and concrete screws. This method considerably enhances the ductility of slab-column connections. Typically, the slab will fail before the complete activation and shear bolts. 3) Enhancing the flexural tension reinforcement by adding an additional layer of concrete or affixing an external reinforcement composed of carbon fibre-reinforced polymers (CFRP) or steel [22-24]. This approach increases the effective depth of the slab, hence increasing its flexural strength; nevertheless, it concurrently leads to increased brittleness in the slab-column connections.

The shear bolt strengthening system with small end anchorages on slab surfaces, an innovative kind of punching shear reinforcement invented by reference [25], was then deeply investigated by reference [19] in the interior slab-column junctions. The references [17-20] conducted several investigations into this kind of strengthening strategy and proved its effectiveness in resisting punching shear in flat slabs. The existing literature lacks a thorough numerical modelling and analysing of shear bolts with two anchorage strategies: small and large anchorages on slab surfaces. However, limited numerical studies conducted by Genikomsou and Polak [26, 27] modelled and analysed reinforced flat slabs with shear bolts incorporating small end anchorages on slab surfaces. They discovered that the best way to model shear bolts is with the beam element type (B32). To avoid concrete failure at anchorages, they also simulated the small end anchorages of shear bolts using assembled beam elements with a size larger than the actual anchorages, provided that they overlap with each other (see Figure 1). However, the current literature still lacks numerical studies related to modelling large anchorages of shear bolts. This approach relies on assumptions that complicate the modelling process, while other simple and reliable techniques have not yet been investigated. Investigating these simpler techniques may provide valuable insights and lead to more precise results. Furthermore, there are a few recent studies concerning the utilisation of diverse configurations of shear bolts, particularly in conjunction with large-end anchorages. All previous studies concentrated on shear bolts with small end anchorages.

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Figure 1. Assembled truss/beam for shear bolt modelling [26]

In this study, the concrete damaged plasticity (CDP) model in ABAQUS is used to numerically calibrate eight interior slab-column junctions shear-reinforced by shear bolts with large and small anchorages to slab surfaces originally tested by reference [28]. The study proposes numerical modelling approaches for representing the end anchorages of shear bolts and assesses the efficacy of the strengthening method for ultimate capacity, failure mode, ductility, stiffness, and energy absorption. Besides, it investigates the various arrangements of shear reinforcements around the supporting region and their impact on the structural integrity. The numerical results are ultimately compared to the applicable design code standards to assess their consistency and identify potential areas for improvement.

3.1 Common modelling parameters for all slabs

Three-dimensional non-linear analysis was conducted using the finite element ABAQUS software on slab-column interior junctions strengthened by shear bolts. The CDP model was utilised to study the non-linear behaviour of concrete slabs, encompassing inelastic deformation, crushing, cracking, and the degradation of stiffness. The damaged concrete plasticity model was previously employed in the analysis of slab SB1 without strengthening [29]; the same procedure is used herein for modelling all slabs. The parameters used for the damaged plasticity model are listed in Table 2.

Table 2. Damaged plasticity parameters used for modelling

The concrete was modelled using 8-node hexahedral brick elements (C3D8R) with reduced integration [30-32]. Both the compression and tension responses of the concrete were modelled with the use of the Hognestad parabola and the stress-crack displacement relationship, respectively. The tensile behaviour was defined by a bilinear tension-softening response, necessitating key concrete parameters, including the maximum tensile strength f'_t and the fracture energy G_f. The parameter G_f denotes the area beneath the tensile stress-crack displacement curve, which is affected by the size of the aggregate and concrete’s quality. The CEB-FIP Model Code 1990 [33] was adopted for estimating G_f as follows:

$G_f=G_{f o}\left(f_{c m} / f_{c m o}\right)^{0.7}$                      (1)

The steel material, including reinforcing flexural bars, shear bolts, nuts, washers, and loading plates, were modelled utilizing truss elements (T3D2). They were defined using the uniaxial stress-plastic strain relationship. Their plastic behaviour was outlined using a data table that included yield stress and the corresponding plastic strain. The embedding technique was used to mimic an ideal bond between reinforcing bars and concrete. Compression and tension damage criteria were ignored due to monotonic loading. The tie constraint was used to delineate the interaction between the surfaces of the loading plate and the concrete. This kind of restriction effectively inhibits movement between two sets of surfaces, despite their differing mesh sizes.

Due to all slabs having identical geometry, a quarter of each was considered for analysis. The test slabs were analysed via quasi-static analysis in ABAQUS. This method necessitates applying a low velocity to guarantee precise outcomes. Thereby, all slab models were displaced at 20 mm/s, which gradually increased according to a smooth amplitude curve from 0 mm/s to 40 mm/s. The slab models were restrained against vertical displacement at their perimeters. Figure 5 depicts the typical geometry and boundary conditions for the slab models used in the simulations. The slab vertical deflection $\Delta$ was measured at the centre of each slab.

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Figure 5. Depicts the typical geometry and boundary conditions for the slab models used in the simulations

A finite element sensitivity analysis was conducted on reference slab model (R) using mesh sizes of 15 mm, 20 mm, and 30 mm. Based on this study, a 20 mm mesh size was selected for the calibration process. This decision was based on the convergence of results and the agreement between finite element and experimental observations particularly in terms of load-deflection responses, cracking patterns and flexural reinforcement behaviour. Figure 6 presents a sensitivity analysis for slab model R for three different mesh sizes (15, 20, and 25).

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Figure 6. Sensitivity analysis for slab model R

3.2 Modelling of shear bolts

The strengthening strategy against punching shear involves using shear bolts with different end anchorages (small, large, and SEs on slab surfaces). Three types of elements in the ABAQUS material library could be used for modelling shear bolts and their anchorages; these elements involve beam, shell, and solid elements. The truss element type was excluded because it resists only compression and tension forces, contrary to the beam element. To study different kinds of modelling approaches for shear bolts, the test slabs (M8S and M8) were selected to represent the slabs with small and large end anchorages, respectively. The goal is only to identify the best element type for shear bolts and their anchorages that would apply to other slabs. For simplicity, a flowchart was designed to illustrate the modelling techniques used for shear bolts, as shown in Figure 7.

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Figure 7. Flowchart of the used modelling strategies of shear bolts for slab models M8S and M8

For shear bolts with small anchorages, four different modelling approaches were considered. Solid elements for shear bolts and anchorages (model S1), beam elements for shear bolts without anchorages (model S2), and beam elements for shear bolts with solid anchorages. In the last kind, the joint between the shear bolt and its anchorages was modelled once constrained (model S3) and once unconstrained (model S4). For the constrained type, the coupling constraint type was used (Ux = Uy = Uz = 0). More details are listed in Table 3 and shown in Figure 8.

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Figure 8. Modelling techniques for shear bolts with small anchorages for slab model M8S

For shear bolts with large anchorages, seven modelling methods were presented. Assembled beam elements for both shear bolts and their anchorages (model L1), two types of sloid element systems for shear bolts and their anchorages (models L2 and L3), two types of shell element systems for shear bolts and their anchorages (models L4 and L5), and beam elements for shear bolts and solid elements anchorage (models L6 and L7). In the last kind, the joint between the shear bolts and their anchorages was modelled once constrained and once unconstrained. More details are listed in Table 3 and shown in Figure 9.

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Figure 9. Modelling techniques for shear bolts with large anchorages for slab model M8

Table 3. Details of modelling strategies for shear bolts and their anchorages

3.3 Optimal modelling strategy for shear bolts

Eleven modelling strategies for shear bolts and their anchorages, both small and large, were proposed and examined. Figure 10 indicates that all proposed strategies demonstrated satisfactory performance in the load-deflection behaviour relative to the experimental slabs (M8S and M8). Additionally, all strategies generated similar crack patterns on the tension surfaces when compared to the corresponding test slabs (M8S and M8). The ultimate failure loads and their corresponding central deflection for all proposed modelling strategies are illustrated in Table 4.

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Figure 10. Numerical load-deflection behaviour for slab models (a) M8S and (b) M8 for various shear bolts modelling strategies

Table 4. Ultimate failure loads and central deflection for all proposed modelling strategies

For the slab M8S (shear bolts with small anchorages), the modelling strategy S4 (beam elements for shear bolts and solid elements for anchorages) showed optimal performance in terms of load-deflection response, crack pattern on the tension surface, straightforward modelling, and less computational time. The other modelling strategies (S1, S2, and S3) demonstrated acceptable results regarding the previously mentioned criteria. However, while modelling strategies S1 and S3 showed acceptable results, they required more time for both modelling and computation. Furthermore, the modelling strategy S2, which uses shear bolts without anchorages, showed results that were below the required level when compared to the other modelling strategies.

Regarding the slab M8 (shear bolts with large anchorages), the model strategy L7 (beam elements for shear bolts and solid elements for anchorages) revealed superior performance in terms of load-deflection response, crack pattern on the tension surface, straightforward modelling, and computational time. Other modelling strategies (L1 to L6) exhibited satisfactory outcomes concerning the aforementioned criteria. Nevertheless, whereas these modelling strategies achieved satisfactory outcomes, they necessitated more time for both modelling and computation.

The modelling strategies S4 and L7 are most effective for modelling shear bolts and their anchorages in slabs strengthened with shear bolts featuring small and large end anchorages, respectively. Consequently, these two modelling strategies are implemented on the remaining strengthened slabs. Figures 11-13 present a comparison of the load-deflection curves of all strengthened slabs for large, small, and SEs, respectively. Figures 14 and 15 depict the crack patterns observed on the tension surfaces of all slab models and the test slabs (M6, M6S, and M6SE). Obvious cracks formed within the enhanced zone for slab models M6, M8a, M6S, M6SE, and M8SE. Consistent with the test findings, these slab models failed in punching mode within enhanced zone [17]. The other slab models failed in the flexural mode linked to punching failure outside the enhanced zone. The consistency between the finite element analysis and the experimental findings regarding load-deflection responses and crack patterns refers to the verified models' ability to evaluate additional parameters.

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Figure 11. Load-deflection curves for strengthened slabs with large anchorages using modelling strategy L7

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Figure 12. Load-deflection curves for all strengthened slabs with small anchorages using modelling strategy S4

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Figure 13. Load-deflection curves for strengthened slabs with SEs using modelling strategy S4

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Figure 15. Top view (left) and bottom view (right) of specimens (a) M6, (b) M8S and (c) M6SE [28]

The slab models (M8-8 mm, M8-10 mm, and M8-12 mm) in Group I demonstrated similar behaviour in terms of load-deflection and failure mode. This suggests that augmenting the anchoring thickness has not affected the punching shear strength. The reason is that the increasing the anchoring thickness kept the critical section unchanged. Nonetheless, increasing the anchoring thickness somewhat improved the punching capacity by about 3% relative to slab model M8-FEA, as seen in Figure 17(a). Increasing the anchorage thickness resulting in increasing the confining pressure between them. Thus, these slab models exhibited a marginal, almost imperceptible enhancement in initial stiffness, ductility, and energy absorption relative to slab model M8-FEA. Moreover, all these slab models recorded ultimate strength higher than the specified flexural strength of the unenhanced slab (V_flex = 328 kN). Hence, all these slabs were failed in flexural mode, which is finally associated with punching failure outside the shear-enhanced region. The crack pattern on the tension surface was similar to that crack pattern formed in slab model M8-FEA, as shown in Figure 14(e).

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Figure 17. Load-deflection behaviour of parametric slab models in a) Group I, b) Group II, and c) Group III

The slab models M8-2, M8-6, M8-7, and M8-8 in Group II exhibited satisfactory outcomes concerning punching capacity and failure mode. The enhancement in punching capacity for these slab models was approximately 3%, 4%, 5%, and 18%, respectively, compared to the slab model M8-FEA, as depicted in Figure 17(b). This is attributed to the supporting zone being well expanded when adopting such arrangements. Additionally, if compared to the other slab models in the same group, the slab model M8-8 recorded an increase in energy absorption and ductility of about 164% and 25%, respectively, over the slab model M8-FEA. This is attributed to the increased confining pressure resulting from the presence of radial placement of shear bolts. Besides, all the slab models in this group exhibited well-defined crack patterns on the tension surfaces, as shown in Figure 18. However, the punching capacity of the other slab models (M8-1, M8-3, M8-4, M8-5, and M8-7) in this group was inferior to that of the slab model M8-FEA. Moreover, all of the slab models in this group recorded a flexural failure mode associated with punching shear outside the shear-enhanced zone except for slab model M8-3, which failed in punching mode beyond the shear-reinforced zone. This failure occurred due to the placement of the shear bolt within a distance equivalent to d = 50 mm (within the critical section). Additionally, the ultimate strength recorded by all slab models exceeded the flexural strength of the unenhanced slab (V_flex = 328 kN), except for slab model M8-3. Moreover, the slab model M8-6 exhibited a better post-failure response compared to other slab models. This phenomenon is attributed to the slight confining pressure applied between anchorages, resulting from the absence of the radial placement of shear bolts.

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Figure 18. Crack patterns on tension surfaces of parametric slab models

The slab models M8S-2, M8S-4 and M8S-7 in Group III demonstrated satisfactory outcomes regarding punching capacity and crack patterns on the tension surfaces. The enhancement in punching capacity for these slab models was on average 7% compared to the slab model M8S-FEA, as depicted in Figure 17(c). These configurations embrace and beyond the critical section around the supporting region, signifying their effectiveness. The other slab models in this group recorded punching strength equal to slab model M8S-FEA, except slab models M8S-5 and M8S-6, which recorded punching strength lower to slab model M8S-FEA. Only the slab model M8S-4 failed in flexural mode, which is finally associated with punching shear outside the shear-reinforced region. Except for slab models M8S-5 and M8S-6, which failed in punching shear within the shear-reinforced zone. Indicating to insufficient amount of shear reinforcement around the supporting zone. The others failed in punching shear mode outside the shear-reinforced zone before attaining the ultimate flexural strength (V_flex = 378 kN). The evidence suggests that these arrangements are unfavourable. Figure 18 displays the crack patterns on the tension surface for all slab models in this group.

The radial arrangement of shear reinforcement had substantial outcomes, especially for the ultimate strength and failure mode, in contrast to other arrangements. The radial arrangement of shear reinforcement provided a better enhancement to the flexural reinforcement, extended the critical section surrounding the support region, and alleviated stress concentration at column corners, particularly in case of adopting shear bolts with large end anchorages.

Shear bolts with large anchorages demonstrated superior mobilisation for flexural reinforcement compared to shear bolts with small anchorages. The large anchorages not only enhanced the flexural reinforcement but also amplified the critical section around the column. Furthermore, the flexural reinforcement commenced activation at a deflection of 2.4 mm across all reinforced slab models.

Small stresses were detected in the shear bolts until the maximum load of the slabs was attained. Only the first bolt commenced activation at a deflection of 10 mm. Typically, the bolts of all slabs began to engage at loads above the ultimate load of the specimen devoid of shear reinforcement (R), since cracking occurred at comparable load levels for all slabs with and without shear reinforcement. The first row of bolts endures more stresses than the following rows because of the shear cracks located near the column.

Finite-element simulations of shear-reinforced concrete flat slabs using shear bolts were conducted using the CDP model to investigate ultimate load, cracking propagation, and central deflection of shear-reinforced interior slab-column junctions. Accurate numerical predictions require proper modelling process, and the calibrated CDP model adopted from Slab SB1 without shear bolts has been shown to reliably capture the structural responses of shear-reinforced slabs. In this study, shear bolts and their end anchorages were represented as beams and solid elements, respectively, and various modelling strategies were explored to simulate their behaviours. Eight flat slabs from previous experimental work were numerically examined in ABAQUS, considering different shear bolt arrangements and end anchorages (small and large). Furthermore, the study evaluated the impact of various arrangements of shear bolts and the thickness of their anchorages on punching resistance by conducting a parametric analysis. This study draws conclusions only from numerical findings, as follows:

1. The numerical calibration results pointed out that the optimal modelling strategy was to use beam elements for shear bolts and solid elements for their anchorages, regardless of whether the anchorages are small or large, and an unconstrained (coupling constraint is overlooked) joint between shear bolts and their anchorages. This method established a dependable baseline for simulation; however, its applicability remains confined to the static loading conditions.

2. The other modelling strategies for shear bolts and their anchorages have shown acceptable representations in terms of load-deflection responses and crack patterns. But these modelling strategies are complex and require a lot of efforts and computational time. Furthermore, using shear bolts solely without anchorages demonstrated unacceptable behaviour compared to other modelling strategies.

3. The majority of the strengthened slab models exhibited a flexural failure mode associated with punching shear beyond the shear-reinforced zone. The probability of such a failure mode escalates with the increasing quantity of dispersed shear bolts around the column. However, the radial arrangement of shear bolts with large anchorages demonstrated enhanced performance for load-deflection responses and deformation capacity.

4. Certain slab models either experienced failure due to punching shear outside the shear-reinforced zone (slab model M8-3) or failed in punching shear inside the shear-reinforced zone (slab models M8S-5 and M8S-6). This suggests that the shear reinforcement was not properly arranged. This highlights a critical area for future refinement in design guidelines.

5. Shear bolts with larger anchorages enhanced the mobilization of flexural reinforcements. The effect is evident in the crack propagation towards the slab edges and corners. This highlights the importance of designing the shear bolts' anchorages for better structural performance.

6. Both the design codes demonstrated conservative predictions; ACI 318 was more conservative than Eurocode 2. However, both of them overestimated punching shear resistance for the slab models M8S-5 and M8S-6. Indicating that these design codes may.

7. Future research needs to expand this analysis to include dynamic loading conditions for assessing performance during seismic or impact events, and to explore a broader spectrum of geometric and material parameters to provide more generalised design recommendations.