FE CFST models stiffened with internal I-steel stiffeners analysed by Al Zand et al. [19].

#### *5.3. Development of the New Analytical Method*

Generally, the cross-sections of steel tubes are classified into Compact, Noncompact and Slender sections based on their ability to buckle under compression stress. The tube's effective-width (*Weff*)-to-thickness (*t*) ratio, usually known as the slenderness ratio (*λ*), is used as a limit for this classification in the majority of related standardised codes. In the current study, the slenderness limits that are specified in ANSI/AISC 360-10 [64] (Chapter I) were adopted for the classification of the rectangular steel tube beams that were filled with concrete (cross-sections of CFST members). Figure 13 presents the relationship between the nominal flexural strength (Nominal moment; *Mn*) and the slenderness ratio (*λ*) of the cross-section of CFST beam's tube. First, the tube's cross-section was classified as a "Compact section" if the *λ* value was within the limits of the compactness ratio (*λp*), which was equal to 2.26 (*Es*/*Fy*) 0.5. Second, if the value of *λ* was larger than that of *λp*, but within

the limits of the noncompactness ratio (*λr*), which was equal to 3.0 (*Es*/*Fy*) 0.5, then the tube's cross-section was classified as a "Noncompact section". Third, when *λ* exceeded the limit of *λ<sup>r</sup>* but was within the limits of the maximum ratio (*λlimit*), which was equal to 5.0 (*Es*/*Fy*) 0.5, then the cross-section of the CFST beam's tube was classified as a "Slender section". This was the case because the ANSI/AISC 360-10 code does not permit the use of Slender CFST beams if their *λ* values exceed the maximum limit (*λlimit*). *Materials* **2021**, *14*, x FOR PEER REVIEW 20 of 26

**Figure 13.** Moment vs. slenderness ratio relationship of the tube's cross-section. **Figure 13.** Moment vs. slenderness ratio relationship of the tube's cross-section.

In this study, a novel analytical method for the prediction of the nominal moment (*Mn*) was developed based on the fundamental theory of stress block diagrams of CFST beams that are internally stiffened with steel stiffeners, as shown in Figure 14. For this purpose, several assumptions were adopted, which are listed as follows: In this study, a novel analytical method for the prediction of the nominal moment (*Mn*) was developed based on the fundamental theory of stress block diagrams of CFST beams that are internally stiffened with steel stiffeners, as shown in Figure 14. For this purpose, several assumptions were adopted, which are listed as follows:

	- ix. Noncompact section (see Figure 14b): this section was assumed to have elastic-plastic behaviour at the tension zone and elastic behaviour at the compression zone, and the steel stress was assumed to be within the limits of *Fy* [12,64]. The concrete ix. Noncompact section (see Figure 14b): this section was assumed to have elastic-plastic behaviour at the tension zone and elastic behaviour at the compression zone, and the steel stress was assumed to be within the limits of *F<sup>y</sup>* [12,64]. The concrete compression

compression stress was assumed to be within the limits of 0.9*fcu* and distributed as a

behaviour, and the steel stress was assumed to be within the limits of *Fy* at the maximum tension face and within the limits of the buckling stress (*Fcr*) at the maximum compression face [12,64]. For this section, a lower concrete compression

stress was assumed, which was taken to be within the limits of 0.8*fcu*.

triangular stress block to the N.A. position.

n ൌM

௬ൌ Weff Weff

tF௬ሺD

tF

 st st

 st st

st

stress was assumed to be within the limits of 0.9*fcu* and distributed as a triangular stress block to the N.A. position.

	- xi. Finally, when the forces over the stiffened CFST beam's cross-section attained equilibrium (see Figure 14), the summarized forms of the new analytical formula for predicting the *M<sup>n</sup>* for each section classification could be expressed as follows: equilibrium (see Figure 14), the summarized forms of the new analytical formula for predicting the *Mn* for each section classification could be expressed as follows:

$$\begin{array}{l} \text{For the Component section } \left(\lambda\_{st} \le \lambda\_p\right) \\ y\_c = \left(2tDF\_y + f\_{cu}W\_{eff}t\right) / \left(4tF\_y + f\_{cu}W\_{eff}\right) \end{array} \tag{2}$$

$$M\_{\rm n} = M\_{\rm p} = W\_{\rm eff} t F\_y (D - t) + t\_{\rm st} d\_{\rm st} F\_y (D - D\_{\rm st}) + t F\_y [y\_c^2 + (D - y\_c)^2] + 0.5 W\_{\rm eff} f\_{\rm mf} (y\_c - t)^2 \tag{3}$$
 
$$\text{For the Noncompact section } (\lambda\_p < \lambda\_{st} \le \lambda\_r)$$

st

For the Compact section (*λst ≤ λp*)

$$y\_n = \left(2tDF\_y + 0.45f\_{cu}W\_{eff}t\right) / \left(4tF\_y + 0.45f\_{cu}W\_{eff}\right) \tag{4}$$

$$M\_{\rm y} = W\_{\rm eff} t F\_{\rm y} (D - t) + t\_{\rm st} D\_{\rm st} F\_{\rm y} \left( D - D\_{\rm st} \right) + t F\_{\rm y} D (D - 2y\_n) + 4 / 3 t F\_{\rm y} y\_n^2 + 0.3 W\_{\rm eff} f\_{\rm cu} (y\_n - t)^2 \tag{5}$$

$$\begin{array}{l} M\_{\text{n}} = M\_{\text{p}} - \left[ \left( M\_{\text{p}} - M\_{\text{y}} \right) \cdot \left( \lambda - \lambda\_{\text{p}} \right) / \left( \lambda\_{\text{r}} - \lambda\_{\text{p}} \right) \right] \\ \text{For the Sleder sec tion } (\lambda\_{\text{r}} < \lambda\_{\text{st}} \le \lambda\_{\text{limit}}) \end{array} \tag{6}$$

$$F\_{\rm cr} = 9\mathcal{E}\_{\rm s} \,/(\mathcal{W}\_{eff}/t)^2\tag{7}$$

eff

cu

eff

cu

(9)

$$y\_s = \left[tDF\_y + \mathcal{W}\_{eff}t\left(0.4f\_{\rm cu} + f\_y - f\_{\rm cr}\right) + t\_{\rm st}d\_{\rm st}\left(F\_y - f\_{\rm cr}\right)\right] / \left(t\left(F\_y + f\_{\rm cr}\right) + 0.4f\_{\rm cu}\mathcal{W}\_{eff}\right) \tag{8}$$

$$\omega = \omega + \omega + \omega = \omega + \omega = \omega + \omega = \omega + \omega = \omega$$

$$\begin{aligned} M\_{\rm ll} &= M\_{\rm cr} = \mathcal{W}\_{eff} t \mathcal{F}\_{\rm cr} (y\_{\rm s} - t/2) + \mathcal{W}\_{eff} t \mathcal{F}\_{\rm y} (D - y\_{\rm s} - t/2) + t\_{\rm st} d\_{\rm st} \mathcal{F}\_{\rm cr} (y\_{\rm s} - d\_{\rm st}/2) \\ &+ t\_{\rm st} d\_{\rm st} \mathcal{F}\_{\rm y} (D - y\_{\rm s} - d\_{\rm st}/2) + 2/3 t \mathcal{F}\_{\rm cr} y\_{\rm s}^2 + 2/3 t \mathcal{F}\_{\rm y} (D - y\_{\rm s})^2 \\ &+ 0.267 \mathcal{W}\_{eff} t\_{\rm cu} (y\_{\rm s} - t)^2 \end{aligned} \tag{9}$$

(**a**)

**Figure 14.** *Cont*.

*Materials* **2021**, *14*, x FOR PEER REVIEW 22 of 26

**Figure 14.** Stress block diagrams of stiffened CFST beam with internal stiffeners: (**a**) compact section; (**b**) non-compact section; (**c**) slender section. **Figure 14.** Stress block diagrams of stiffened CFST beam with internal stiffeners: (**a**) compact section; (**b**) non-compact section; (**c**) slender section.

Accordingly, the embedded steel stiffeners that were provided to stiffen the sections of the CFST beams/columns significantly reduced the flat distance of their tube's flanges/walls, which led to a delay in the tube's buckling failure, as discussed earlier in this paper and previously confirmed in the literature [19,22,47]. On that basis, the classification of the tube sections of the CFST beams could be changed from Slender to Noncompact and/or to Compact due to the influence of these stiffeners, as evidenced in the comparisons between the stiffened slenderness ratios (*λst*) and the *λ* values in Table 4. Furthermore, when compared to the *Mu* values obtained from the current study and the additional models analysed in [19], the new analytical method achieved the best prediction values (*Mn*), with MV and COV values of 0.831 and 0.049, respectively, as compared to the existing theoretical methods shown in Table 4. Accordingly, the embedded steel stiffeners that were provided to stiffen the sections of the CFST beams/columns significantly reduced the flat distance of their tube's flanges/walls, which led to a delay in the tube's buckling failure, as discussed earlier in this paper and previously confirmed in the literature [19,22,47]. On that basis, the classification of the tube sections of the CFST beams could be changed from Slender to Noncompact and/or to Compact due to the influence of these stiffeners, as evidenced in the comparisons between the stiffened slenderness ratios (*λst*) and the *λ* values in Table 4. Furthermore, when compared to the *M<sup>u</sup>* values obtained from the current study and the additional models analysed in [19], the new analytical method achieved the best prediction values (*Mn*), with MV and COV values of 0.831 and 0.049, respectively, as compared to the existing theoretical methods shown in Table 4.

#### **6. Conclusions 6. Conclusions**

The conclusions of the investigated CFST beams are summarized as follows: The conclusions of the investigated CFST beams are summarized as follows:

 The experimental investigation confirmed that the bending capacity of the suggested prefabricated Slender CFST beams made from two pieces of C-sections was enhanced by about 3.7 times even when filled with 70% replacement recycled concrete material.


It is worth highlighting the main research limitation of the current investigation, namely that the uncertainties related to this model could be further examined as they are very significant in terms of the parameters of structural behavior. In addition, further experimental/numerical investigations, other than the rectangular cross-sections under static/dynamic loading scenarios, could be conducted on the stiffened CFST beams.

**Author Contributions:** Conceptualization, A.W.A.Z., R.A.-A. and W.H.W.B.; Data curation, A.W.A.Z., M.M.A. and E.H.; Formal analysis, A.W.A.Z., M.M.A. and Z.M.Y.; Funding acquisition, A.W.A.Z.; Investigation, M.M.A. and E.H.; Methodology, R.A.-A.; Project administration, A.W.A.Z., R.A.-A., W.H.W.B. and Z.M.Y.; Resources, W.H.W.B. and E.H.; Software, A.W.A.Z. and W.M.T.; Supervision, W.H.W.B.; Validation, A.W.A.Z., M.M.A., R.A.-A. and Z.M.Y.; Visualization, W.M.T., E.H. and Z.M.Y.; Writing—original draft, A.W.A.Z., M.M.A., W.M.T. and E.H.; Writing—review and editing, A.W.A.Z., W.H.W.B., W.M.T. and Z.M.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Universiti Kebangsaan Malaysia, grant number GGPM-2020-001.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data are presented in the article.

**Conflicts of Interest:** The authors declare no conflict of interest.
