*3.2. Efficiency of Mesh Reinforcement*

Undoubtedly, the structural suitability of materials with compressive strength below 8MPa is limited to structures of secondary importance such as partition walls, leveling layers, etc. The authors' experience shows that foamed concretes with such densities are increasingly used in load-bearing elements such as foundations, structural walls or ceilings. Usually these are composite structures with a thin concrete slab or reinforced with cores. The introduction of reinforcement should simplify such structures. The second effect of the reinforcement may be the reduction of susceptibility of usually fragile foamed geopolymer to accidental cracking.

The main objective of this part of research was to evaluate the effectiveness of reinforcement with glass fiber mesh. The results of these tests are presented in Tables 7 and 8, which show the magnitude of the failure force in the bending test and the corresponding deflection, with a description of the failure mode. Three typical modes of failure were observed: rupturing of mesh fibers, delamination of the mesh, and crushing of concrete in the compressed zone, which was usually preceded by the development of flexural cracks. Examples of these are shown in Figure 7.


**Table 7.** Test results for beams reinforced at the bottom.

**Table 8.** Test results for internally reinforced beams.


The use of reinforcement should increase the load-bearing capacity. It is clear that the reinforcement fulfilled this requirement and that the carrying capacity of the reinforced samples was greater than for models without reinforcement. The best results were obtained for the samples with the highest density, which is in line with expectations. For example, a 43% increase in flexural capacity was obtained for the T\_1%\_in model, and a 90% increase for the B\_1%\_bt model. It should be noted that all of the samples were reinforced with the same fiberglass mesh, so the greatest strengthening effects could therefore be expected for models made of the weakest material, i.e., those with the lowest density. A graphical comparison of the failure forces in the bending test is shown in the diagram in Figure 8.

**Figure 7.** Observed failure modes: (**a**) rupture of the bottom fiber mesh; (**b**) delamination of the bottom fiber mesh; (**c**) rupture of the fiber mesh; (**d**) simultaneous crushing of concrete and fiber rupture.

**Figure 8.** Comparison of failure forces in the bending test.

Theoretically, the flexural capacity of a concrete section is most strongly influenced by the reinforcement strength and the arm of the internal forces, and the mechanical properties of the concrete itself are of secondary importance. For the models tested here, the change in load capacity after using a more foamed geopolymer was not expected to drop by more than about 12%. This hypothesis was confirmed only for the internally reinforced samples based on fly ashes from the Jaworzno power plant (J\_1%\_in, J\_2%\_in, and J\_3%\_in). Of the models made of C-type ash, only the T\_1%\_in and T\_1%\_bt models had a load capacity that was comparable to those based on Jaworzno fly ash.

Despite the larger arm of the internal force, the load capacity for most samples reinforced with mesh attached along the bottom surface was lower. This phenomenon can be easily explained. An analysis of the failure modes (summarized in Table 7 and illustrated in Figure 7) shows that only the densest geopolymers (foamed with 1% H2O2) provided a sufficient bond. Models foamed with additions of 2% and 3% of the foaming agent mostly broke as a result of debonding of the fiberglass mesh. The presence of pores reduces the adhesion surface, creating a poorer bond to the composite fibers, and this led to premature failure before the strength of glass fibers was exceeded. The cause of the early failure of the internally reinforced samples was different; in general, these failed after the rupture of composite fibers (Figure 7c). An inspection of the broken sample showed increased brittleness of the composite (Figure 9). The most probable reason for this is corrosion

of the fiberglass caused by the alkalinity of the geopolymer. Although the tests of the samples showed a similar pH of below 11.5 for all samples, this may exceed 13.5 for a fresh mix [57]. This effect was magnified in a fresh mixture by the presence of an unreacted sodium hydroxide activator [58,59]. In the studies carried out here, all the geopolymers were fabricated using the same ratio of activator to precursor. In lignite coal fly ashes, which are poorer in aluminum and silicon, the consumption of sodium hydroxide in the reaction mechanism (geopolymer synthesis) may be slower, thus extending the exposure time of the glass mesh to the alkaline solution and causing increased brittleness of the fibers, as shown by the samples based on the Belchatow and Turow fly ashes.

**Figure 9.** Brittle rupture of fiberglass in which the visible breakthrough does not contain free fibers.

Most of the internally reinforced beams broke at deflections of between 0.5 and 1 mm, with the exceptions being the T\_1%\_bt and T\_2%\_in beams, which failed at a deflection of around 0.15 mm (Tables 7 and 8). The reason for this specific behavior can be found by analyzing the load-deflection relationship, as shown in Figure 10. For most beams, there is an almost linear increase in the deflection in the first phase until cracking occurs (Figure 10a). There is then a rapid increase in the deflection, associated with the development of the crack, until the composite takes all the internal tensile force. We then see a further increase in the transferred load, until the fibers break. The cracking in both beams shown in Figure 10 occurred at a load of approximately 0.34 kN, and the sample shown in Figure 10b failed shortly after cracking. Premature rupture of the brittle glass fibers of the T\_2%\_in sample did not allow for the increase in the bearing capacity characterizing the post-crack phase, as described above. A similar effect was caused by composite delamination in the T\_1%\_bt sample. An increase in the brittleness of the fiberglass was noticed in all beams made of geopolymer synthesized from C-type ashes (Turow and Belchatow); however, strongly foamed material allowed for greater deflection due to crushing of the compressed zone (Figure 7d).

#### *3.3. Analytical Model*

Determination of the ultimate moment of resistance of composite-strengthened geopolymer concrete is based on similar assumptions to the calculations for steel-reinforced concrete elements. The most important are: (i) the cross-section remains a plane after deformation; (ii) the strains in composite fiber and surrounding concrete are compatible; and (iii) the tensile strength of the concrete can be ignored. Of course, due to its specific material properties, an appropriate strain limit should be assumed for geopolymers. Likewise, the stress-strain relationship must be based on a suitable function. Figure 11 illustrates the assumptions made in the model described here.

**Figure 10.** Loading force-deflection curves: (**a**) typical behavior of an internally reinforced beam; (**b**) premature failure caused by brittle fiber rupture or fiberglass debonding.

**Figure 11.** Model of a fiberglass reinforced beam under flexure.

The bending resistance of a cross-section can be determined based on the equilibrium of internal forces, according to the following Equations (2) and (3):

$$M\_E = 0.25 F\_E \cdot l\_b = F\_f (h - \delta\_\mathbb{G} \ge \mathfrak{x}),\tag{2}$$

$$
\Psi \mathfrak{e}\_c E\_c b \mathfrak{x} = F\_{f\_{\prime}} \tag{3}
$$

where:

*Ff*—fiber breaking force,

*lb* = 100 mm (the span of beam),

*ε<sup>f</sup>* , *εc*—strains of the composite fiber and the outmost geopolymer fiber, respectively, *Ef* , *Ec*—elasticity moduli for the composite fiber and concrete [MPa], respectively.

The lower limit of the cross-section bearing capacity is the cracking moment:

$$M\_{\rm cr} = f\_{\rm ct} bh^2 / 6. \tag{4}$$

where *fct* is the tensile strength of foamed geopolymer.

The plain section remains plain, therefore:

$$
\varepsilon\_{\mathfrak{c}} = \varepsilon\_f \frac{\mathfrak{x}}{d - \mathfrak{x}} = \frac{F\_f}{E\_f A\_f} \frac{\mathfrak{x}}{d - \mathfrak{x}}.\tag{5}
$$

As shown in Figures 4–6, the stress-strain relationship can be expressed by the parabolic function in (1). In this case, the area of the compressive zone is characterized by the parameter *ψ* and the location of its center of gravity by the parameter *δ<sup>G</sup>* (Figure 11). The values of these parameters change with an increase in the strain, reaching a maximum at the ultimate strain. For a linear stress-strain relationship (triangular), *ψ* = 0.5, and *δ<sup>G</sup>* = <sup>1</sup> <sup>3</sup> . Table 9 shows the values of the coefficients *ψ* and *δ<sup>G</sup>* for the geopolymers tested here, calculated for the parabolic relationship in (1) based on the assumption that the ultimate deformation of the compressed zone is reached, as shown in Table 5. The dependence of *ψ* and *δ<sup>G</sup>* on the deformation significantly complicates the calculations. The parameters presented in Table 9 refer to the situation in which the ultimate strain is achieved in the compressed concrete; for lower strains, these parameters will be lower, aiming at the abovementioned values characterizing the linear relationship. The adoption of a simplified, triangular model over the entire range of deformation leads to a slight underestimate of the load capacity, which can be neglected. This is evidenced in the last column of Table 9, which shows the ratio between the capacity calculated for the triangular relationship and the capacity calculated for the parabolic Relationship (1).


**Table 9.** Parameters for the compressed zone of the foamed geopolymer and the capacity reduction ratio for a simplified stress-strain model.

For the "bt" specimens, i.e., those strengthened along the bottom surface, the force that breaks the composite fibers should not exceed the bond resistance *Fb*. To reflect this phenomenon, the end anchorage model was adopted (given in the *fib* bulletin 90 [60]). When the distance to the first crack *lb* is shorter than the required anchorage length, the debonding force should be limited using the formula proposed by Chen and Teng [61]:

$$F\_b(l\_b) = F\_b \beta\_l = F\_b \sin\left(\frac{\pi}{2} \frac{l\_b}{l\_{b,\text{lim}}}\right). \tag{6}$$

The bond force for a fully anchored composite is equal to:

$$F\_b = 0.25b\sqrt{2E\_f \mathbf{t}\_f f\_c^{2/3}},\tag{7}$$

and the maximum bond length, according to [60], is:

$$l\_{b, \text{lim}} = 0.9 \pi \sqrt{\frac{E\_f t\_f}{8 f\_c^{2/3}}},\tag{8}$$

where:

*tf*—thickness of composite (mm),

*fc*—compressive strength of foamed geopolymer (MPa).

The results of a theoretical analysis of the tested beams are presented in Table 10. Figure 12 shows a comparison of these theoretical results with the values obtained from laboratory testing, expressed as a ratio of the empirical laboratory value to the result predicted using the above formulas.


**Table 10.** Comparison of laboratory determined and theoretically predicted failure forces.

**Figure 12.** Ratios of laboratory determined/predicted flexural capacities of tested beams.

An analysis of the values listed in Table 10 shows that the delamination force is greater than the fiber breaking force for the GFRP composite used here, for all models, a result that is inconsistent with the observed modes of failure (listed in Table 7). All models foamed by the addition of hydrogen peroxide >1% failed due to delamination, indicating that the delamination model adopted here significantly overestimates the results. In future, it will be necessary to develop a model that is suitable for porous materials. To the best of the authors' knowledge, such a model has not yet been developed for externally reinforced expanded concretes.

The second reason for overestimating the load capacity in the case of internally reinforced models is the increased brittleness of the glass fibers as a result of accelerated corrosion in an alkaline environment, as described above.

The probability of premature delamination increases with the degree of foaming. The decreasing strength of the foamed geopolymer may be identified as the main factor promoting debonding. When analyzing most of the methods of predicting the debonding strength, a strong correlation between fracture energy and substrate strength can be seen [62]. For this reason, the Formula (6) can be corrected by a factor that depends on the strength of the tested geopolymer. Figure 13 shows the effect of the geopolymer strength on the

relative error of the debonding force estimation (expressed as the ratio of the tensile force of the composite fiber corresponding to the laboratory-obtained bending resistance and the debonding force calculated according to Expression (6)).

**Figure 13.** Effect of the compressive strength on the relative error of the adhesion force estimation.

The analysis of the error in estimating the debonding force enclosed the samples for which this form of failure was observed (Table 7). Relative error can be estimated with the use of the trend function shown in Figure 13. The value of this function expresses the correction coefficient of the bond force. Introducing it into (7), the modified formula for the bond to the foamed geopolymer can be obtained:

$$F\_b = 0.2439 f\_c^{-0.505} 0.25 b \sqrt{2 E\_f t\_f f\_c^{2/3}} \approx 0.0625 b \sqrt{2 E\_f t\_f f\_c^{-1/3}}.\tag{9}$$

To find the correction factor for the corrosion aging effect, the laboratory determined and predicted flexural capacities were compared. The research showed faster corrosion in geopolymers based on lignite coal fly ash, therefore the analysis was carried out separately for samples made of ashes from Jaworzno and samples made of ashes from Turow and Belchatow. Figure 14 shows the effect of content of foaming agent on the aging factor expressed as a ratio of flexural capacity determined in laboratory and theoretically predicted for ultimate fiber breaking force (listed in Table 10).

Given in Figure 14 trend functions express the estimation error and at the same time the correction factor for to the corrosion aging effect. For the anthracite-coal-fly-ash based foamed geopolymers the fiber breaking force *Ff* given in the Formulas (1) and (2) can be reduced according to the Equation (10):

$$F\_{ff\,\%} = 0.8F\_f \tag{10}$$

Consequently, for the lignite-coal-fly-ash based foamed geopolymers, the reduced fiber breaking force *Fffg* is equal:

$$F\_{f\%} = (0.8 - 0.15 D\_{fa}) F\_f \tag{11}$$

where *Df a* is the content of foaming agent in the geopolymer mixture (given in %).

To check the correctness of proposed empirically modified method, the theoretical flexural capacity of the tested samples was recalculated using the presented formulas. The results of these analyzes are shown in Figure 15. The comparison of the graphs in

Figures 12 and 15 shows a significant improvement in the accuracy of the load-bearing capacity estimation. The correlation coefficient for modified model is 0.835.

**Figure 14.** Effect of the content of foaming agent on the aging effect of glass fibers.

**Figure 15.** Ratios of laboratory determined/predicted flexural capacities of tested beams for empirically modified model.
