### Effect of Wall Thickness

Fig. 5 compares the distribution of the longitudinal Mises stress on the windward and leeward sides of the tower in accordance with the wall thickness (i.e., towers 200-300WTT, 140-210WTT, and 100-150WTT with a fixed wall thickness ratio of 2:3). On the windward side, the similar patterns of stress distribution are observed in towers 140-210WTT and 200-300WTT, while the stress distribution in tower 100-150WTT deviates from the other two cases because of its lower stiffness (Fig. 5a). Because the tapered section decreased the inertia of the moment along the tower height, the tensile stress increased linearly along the tower height until 40 m, but it starts decreasing from 40 m height due to the effect of prestressing force. The tensile stress along the tower height in towers 140-210WTT and 200-300WTT nearly vanishes at around 100 m, and the compressive stress rapidly increased. In tower 100-150WTT, the boundary of the tension and compression zones lies around 60 m in height. The maximum tensile stress on the windward side was 5.25 MPa, which is within the range of tensile strength of the used UHPC material. Thus, the tensile crack would not occur on this side.

As shown in Fig. 5b, only compressive stress developed on the leeward side of the towers. The compressive stress increases linearly along with the height until 40 m along the tower height. The stress distribution between 40 and 100 m was relatively smooth. For towers 200-300WTT and 140-210WTT, the maximum compressive stress of about 13 and 20 MPa occurred in this region, respectively. The maximum compressive stress in tower 100-150WTT was about 30 MPa at the height of 80 m. The tower structure was in a compression state with its maximum value of about 30 MPa, which was significantly less than the compression strength of UHPC material. Thus, the compression failure would not occur in this test case. As the tower thickness increased, the tensile and/or compressive stress decreased.

Fig. 6 compares the lateral displacement of three tower models along with height. The maximum displacement occurred at the top for all tower models. The profiles of the lateral displacement depict the flexural behavior of the towers. As the wall thickness increased, the lateral displacement of the tower decreased. The lateral displacement was relatively small from the base to 50 m height in all the tower models, and increased significantly beyond the height of 50 m. Such result is attributed to the nonuniform load and stiffness along the height. The allowable deformation at the top of the tower is typically about 0.5–0.8% of the tower height, approximately 600–960 mm. The analysis results showed that the maximum lateral displacement of towers 100-150WTT, 140-210WTT, 200-300WWT were 855.1, 637.8, 461.4 mm at tower top, respectively, which satisfied the deformation requirement.

Fig. 7 shows the non-dimensionalized tensile stress (*σ*/*f*_{Utk}) and the non-dimensional lateral displacement (*u*/*u*_{max}) on the windward side of the tower with various wall thicknesses, In this figures, *σ* is the maximum compressive stress; *f*_{Utk} is the specified tensile strength of UHPC (= 6 MPa for UHPC-200); *u* is the lateral displacement at the tower top; and the *u*_{max} is the allowable lateral displacement at the tower top in design code (GB50135, 2019). Based on the computed results, the regression curves of the tensile stress and lateral displacement are established as a function of the wall thickness in the tower with the wall thickness ratio of 2:3:

$$\frac{\sigma }{{f_{{{\text{Utk}}}} }} = 1.725 - 0.0056t_{{\text{w}}} \le 0.933,$$

(1)

$$\frac{u}{{u_{\max } }} = 0.3356 - 0.0011t_{w} ,$$

(2)

where *t*_{w} is the wall thickness in mm (= 100 to 200 mm in this study); and *u*_{max} is the maximal allowable lateral displacement (= 960 mm in this study).

### Effect of Wall Thickness Ratio

To investigate the effect of wall thickness ratio, three tower models with different thickness ratios, 100-200WTT, 200-300WTT, and 200-240WTT, are chosen. Their wall thickness ratio is 1:2, 2:3, and 5:6, respectively. Fig. 8 compares the distribution of the longitudinal Mises stress on the windward and leeward sides of the towers.

On the windward side, the tensile stress linearly increased along with the height from the base to 40 m (Fig. 8a). In towers 200-300WTT and 200-240WTT, the tensile stress slightly decreased in the range of 40–80 m, and it became compressive stress at the height of about 80 m. In tower 100-200WTT, the maximum tensile stress was greater than 5 MPa, and the stress became compression at the height of 60 m. As the wall thickness ratio increased from 1:2 to 2:3, the tensile stress distribution at the windward side did not change significantly. However, when the wall thickness ratio increased from 2:3 to 5:6, the tensile stress decreased by 2.06 MPa at 40 m height.

As shown in Fig. 8b, the stress distribution on the leeward side was in compression through the height regardless of the wall thickness ratio. The compressive stress increased gradually until the height of 20 m, and rapidly increased between 20 and 40 m, particularly in tower 100-200WTT. The stress distribution was relatively smooth between 40 and 100 m where stress was governed by axial force rather than flexural moment. The maximum compressive stress occurred at the height of 60 m, which was less than 20 MPa. As the wall thickness ratio increased, the compressive stress at the leeward side decreased. The discrepancy of the maximum compressive stress between 1:2 and 5:6 wall thickness ratios was 5.03 MPa. This is because the smaller wall thickness decreases the stiffness, which increases the compressive stress.

Fig. 9 shows the lateral displacement along the height of the tower with various wall thickness ratios. The lateral displacement was relatively small below 50 m in height, and significantly increased in the upper region of the tower. The displacement curves of towers 200-300WTT and 200-240WTT were almost overlapping, and the maximum displacement was 461.4 and 450.9 mm, respectively, which satisfied the maximum allowable displacement of 960 mm. The displacement in tower 100–200 WTT was greater than that of the other towers, showing the maximum value of 830.4 mm, which was less than the allowable maximum deformation (i.e., 0.5–0.8% of the total height = 600–960 mm). It is noted that the tower wall thickness ratio does not significantly affect the displacement below 50 m, but it impacts on the top displacement of the towers.

Fig. 10 shows the non-dimensionalized tensile stress (*σ*/*f*_{Utk}) and the non-dimensional lateral displacement (*u*/*u*_{max}) on the windward side of the tower for various wall thickness ratios. Based on the computed results, the regression curves of the tensile stress and lateral displacement were established as a linear function of the wall thickness ratio:

$$\frac{\sigma }{{f_{{{\text{Utk}}}} }} = 1.79 - 1.76R_{w} ,$$

(3)

$$\frac{u}{{u_{\max } }} = 0.49 - 0.55R_{w} \ge 0.12,$$

(4)

where R_{w} is the wall thickness ratio (= 0.5 to 0.83 in this study).

### Effect of Prestressing Tendon

Figs. 11, 12 show the effect of prestressing force on the distributions of the longitudinal stress and lateral displacement in tower 200-300WTT that exhibits relatively low stress distribution and displacement. As shown in Fig. 11a, the maximum tensile stress in the tower without prestressing force was 12.4 MPa under the rated wind load, which was greater than the specified tensile strength of UHPC (i.e., 6 MPa). The prestressing force decreased the maximum tensile stress by more than 50%, while it increased the maximum compressive stress simultaneously. This result demonstrates that the prestressing force reduces the tensile stress effectively, which consequently improves the cracking resistance of the hybrid tower.

Fig. 12 compares the lateral displacement distribution in the towers with and without prestressing force. The lateral displacement was relatively small until the height of 50 m, and reached the maximum value of 461.8 mm at the height of 118 m. Regardless of the prestressing force, the displacement distribution was almost the same because the prestressing force contributed to shift the stress distribution but did not affect the cross-sectional stiffness. This result indicates that the effect of prestressing force on the lateral displacement is negligible because the lateral displacement is significantly affected by the cross-sectional stiffness. Thus, the prestressing force level needs to be determined from the section stress requirement.

To estimate the effect of the prestressing axial compression on the tensile stress of the section, a generic parameter *t*_{w}*R*_{w}*n* is introduced, in which, the axial compression ratio (*n*) is defined as follows:

$$n = \frac{{f_{py} A_{ps} }}{{f_{Uc} A_{U} }},$$

(5)

where *f*_{py} is the yield strength of prestressing tendon; *A*_{ps} is the total cross-sectional area of the prestressing tendon; *f*_{Uc} is the design strength of UHPC (= 95 MPa for UHPC-200); and *A*_{U} is the cross-sectional area corresponding to the maximum tensile stress zone (= 5.53 m^{2} for 200-300WTT with prestressing) at the height of 40–50 m (refer to Fig. 11a).

Fig. 13 shows the non-dimensionalized tensile stress (*σ*/*f*_{Utk}) in accordance with the wall thickness (*t*_{w}) in mm multiplied by wall thickness ratio (*R*_{w}) and axial compression ratio (*n*), and the non-dimensional lateral displacement (*u*/*u*_{max}) in accordance with the wall thickness (*t*_{w}) in mm multiplied by wall thickness ratio (*R*_{w}) as defined in Eqs. (6, 7), respectively. When the *t*_{w}*R*_{w}*n* is greater than 1.58, *σ*/*f*_{Utk} is significantly decreased to 0.33 (Fig. 13a). Similarly, when *t*_{w}*R*_{w} is greater than 93.3, *u*/*u*_{max} significantly decreased to 0.12. It is noted that Eqs. (6, 7) can be used in the value of *t*_{w}*R*_{w}*n* from 0.85 to 2.82 and in *t*_{w}*R*_{w} from 50 to 167, respectively, in this study:

$$\frac{\sigma }{{f_{{{\text{Utk}}}} }} = 1.7 - 0.49t_{{\text{w}}} R_{{\text{w}}} n \le 0.933,$$

(6)

$$\frac{u}{{u_{\max } }} = 0.37 - 0.002t_{{\text{w}}} R_{{\text{w}}} \le 0.23.$$

(7)

The approximation of the tensile stress in Eq. (6) can be used to check the tension limit as follows:

$$\frac{\sigma }{{f_{Utk} }} = 1.7 - 0.49t_{w} R_{w} n \le \frac{{\sigma_{allow} }}{{f_{Utk} }},$$

(8)

where σ_{allow} is the allowable tensile stress. From the above relationship, the minimum requirement for *n* is obtained:

$$n = \frac{{f_{py} A_{ps} }}{{f_{Uc} A_{U} }} = \frac{{f_{py} }}{{f_{Uc} }}\rho_{ps} \ge \frac{{3.47 - 2.04\sigma_{allow} /f_{Utk} }}{{t_{w} R_{w} }}.$$

(9)

Finally, the minimum requirement for the prestressing–reinforcement ratio can be calculated by considering the allowable tensile stress:

$$\rho_{ps,min} = \frac{{3.47 - 2.04\sigma_{allow} /f_{Utk} }}{{t_{w} R_{w} }}\left( {\frac{{f_{Uc} }}{{f_{py} }}} \right).$$

(10)