Research review on demountable shear connectors for prefabricated steel-concrete composite beams
Article Text (Baidu Translation)
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摘要: 为响应国家“双碳”战略对交通基础设施绿色可持续发展的要求, 针对装配式钢-混组合梁可拆卸剪力连接关键技术开展了系统研究, 通过模块化设计与装配式施工, 有效解决了传统焊接剪力连接组合梁资源回收难题；基于近十年国内外相关研究成果的梳理, 重点分析了四大类可拆卸剪力连接件(包括高强螺栓型、盲栓型、可拆卸压型钢板型和创新可拆卸型)的构造特征与力学性能；通过静力与疲劳荷载作用可拆卸连接件推出试验和梁式试验, 揭示了不同连接形式的滑移演化规律、破坏模式以及承载机理。试验结果表明：单螺母螺栓连接件极限抗剪强度达传统焊钉的95%, 贯穿式螺栓连接件疲劳性能显著提升；而创新可拆卸型连接件展现出优异的长期性能, 但普遍存在加工精度要求高、拆卸工艺复杂等技术瓶颈；基于现行设计规范, 提出的涵盖混凝土压溃、螺栓剪切和复合失效模式的抗剪承载力、抗剪刚度计算公式, 通过修正相关系数显著提升计算精度；未来需进一步优化连接件构造以突破成本与工艺限制, 明确复杂服役环境下长期性能演变规律, 建立全寿命周期性能预测模型, 完善相关设计施工规范。研究结果为促进可拆卸连接件技术的工程应用以及桥梁工程装配化建造和可持续发展提供了理论基础和技术支撑。Abstract: In response to the national "dual carbon" strategy for green and sustainable development of transportation infrastructure, a systematic study was conducted on the key technologies for demountable shear connections in prefabricated steel-concrete composite beams. Modular design and prefabricated construction were adopted to effectively address the challenges of resource recycling in traditional composite girders with welded shear connections. Based on a review of relevant research achievements over the past decade, the structural characteristics and mechanical performance of four categories of demountable shear connectors were particularly analyzed, including high-strength bolted, blind-bolted, detachable profiled sheeting, and innovative demountable types. Static and fatigue tests on demountable connectors via push-out and beam tests were conducted to reveal the slip evolution, failure modes, and load-bearing mechanisms under different connection configurations. Test results indicate that the ultimate shear strength of single-nut bolted connectors reaches 95% of that of conventional welded studs, while through-bolted connectors exhibit significantly enhanced fatigue performance. Innovative demountable connectors demonstrate excellent long-term performance, but generally face technical challenges such as high manufacturing precision requirements and complex disassembly processes. Based on current design specifications, proposed calculation formulas for shear capacity and shear stiffness considering concrete crushing, bolt shear, and composite failure modes. The accuracy of the predictions is notably improved by modifying relevant coefficients. Future efforts should be made to optimize connector details to overcome cost and process limitations, clarify long-term performance under complex service conditions, establish life-cycle performance prediction models, and refine relevant design and construction specifications. The research findings provide theoretical and technical support for promoting the engineering application of demountable connectors and advancing the prefabrication and sustainable development of bridge engineering.
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图 14 可拆卸剪力连接件疲劳试验S-N曲线
Figure 14. Fatigue test S-N curves of demountable shear connectors
图 15 可拆卸装配式组合梁抗弯试验
Figure 15. Flexural test of composite beams with demountable connectors
表 1 埋入式螺栓连接件推出试验汇总
Table 1. Summary of push-out tests with embedded bolt connectors
文献来源 年份 连接件类型 设计要素 连接件排列方式 螺栓类型 公称直径/mm 混凝土强度 [18] 2010 双螺母螺栓连接件 1行×1列 ASTM325 22 C20 [9] 2014 单螺母螺栓连接件 2行×2列 国标8.8 16 C30 1行×2列 国标8.8 24 C30 [22] 2015 焊钉螺栓连接件 1行×2列 ML15 18 C30 [12] 2019 单螺母螺栓连接件 2行×2列 Duplex2205 19 C35 [21] 2022 T型螺母螺栓连接件 2行×2列 国标8.8 20 C60 [20] 2023 单螺母螺栓连接件 1行×2列 国标8.8 22 C60 [26] 2023 新型不锈钢螺栓连接件 1行×2列 Austenitic2-70 12 C50 
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表 3 盲栓连接件推出试验汇总
Table 3. Summary of push-out tests with blind bolt connectors
文献来源 年份 连接件类型 设计要素 连接件排列方式 螺栓类型 公称直径/mm 混凝土强度 [18] 2010 Hollo型盲栓连接件 1行×1列 ASTM325 22 C20 [100] 2015 Hollo型盲栓连接件 2行×1列 国标10.9 20 C35 Ajax型盲栓连接件 2行×1列 国标9.8 20 C35 焊钉连接件 2行×1列 国标6.8 19 C35 [42] 2015 焊钉连接件 2行×1列 国标8.8 20 C45 Hollo型盲栓连接件 2行×1列 国标8.8 20 C45 Ajax型盲栓连接件 2行×1列 国标8.8 20 C45 [39] 2020 Ajax型盲栓连接件 1行×1列 国标8.8 20 C35 [101] 2021 Hollo型盲栓连接件 2行×1列 国标8.8 20 钢FRP [41] 2021 Ajax型盲栓连接件 1行×2列 国标8.8 20 钢FRP Lindapter型盲栓 1行×2列 国标8.8 20 钢FRP Injected SRR型盲栓 1行×2列 国标8.8 20 钢FRP
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表 4 压型钢板可拆卸连接件推出试验汇总
Table 4. Summary of push-out tests with profiled steel decking demountable shear connectors
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表 5 新型可拆卸连接件推出试验汇总
Table 5. Summary of push-out tests with novel demountable connectors
文献来源 年份 连接件类型 设计要素 连接件排列方式 螺栓类型 公称直径/mm 混凝土强度 [57] 2018 新型双螺栓连接件 1行×2列 国标8.8 22 C50 [68] 2018 高强螺栓锥形抗剪连接件 单侧单个布置 国标10.9 16 C40 [64] 2018 新型可拆卸开孔钢管连接件 单侧单个布置 20 C55/灌浆C50 [90] 2019 螺旋弹簧连接件 2行×1列 20 C40 [91] 2020 机械耦合器-螺栓剪力连接件 1行×2列 国标10.9 16 C40 1行×2列 国标10.9 20 C40 [78] 2021 新型可拆卸螺栓-焊接混合剪力连接件 2行×2列 国标8.8 20 C30/灌浆C45 [81] 2022 可拆卸式锁紧螺栓剪力连接件 1行×2列 国标8.8 20 C50 [85] 2024 新型型钢-螺栓连接件 单侧单个布置 国标8.8 22 C40
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表 6 埋入式螺栓连接件组合梁弯曲试验汇总
Table 6. Summary of bending tests on composite beams with embedded bolt connectors
文献来源 年份 连接件类型 设计要素 跨径/m 加载 混凝土 连接件/mm 钢梁/mm [123] 2020 单螺母螺栓连接件 4.0 正弯矩四点 C30 20行×2列/国标8.8/公称直径20 工字钢Q235/HM294×200×8×12 [25] 2022 焊钉螺栓连接件 1.6 负弯矩四点 UHPC 10行×5列/国标4.8/公称直径16 钢板Q345/厚度10 [124] 2022 单螺母螺栓连接件 3.0 负弯矩三点 C40 17行×2列/国标8.8/公称直径16 工字钢Q235/HM294×200×8×12 [125] 2024 单螺母螺栓连接件 2.9 负弯矩四点 UHPC 18行×2列/国标8.8/公称直径16 直/波形腹板Q235/HM200×200×15×8
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表 7 贯穿式螺栓连接件组合梁弯曲试验汇总
Table 7. Summary of bending tests on composite beams with through-bolt connectors
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表 8 盲栓连接件组合梁弯曲试验汇总
Table 8. Summary of bending tests on composite beams with blind bolt connectors
文献来源 年份 连接件类型 设计要素 跨径/m 加载 混凝土 连接件/mm 钢梁/mm [42] 2015 Hollo型盲栓连接件 6 正弯矩四点 C45 27行×1列/国标8.8/公称直径20 工字钢S355/UB460×190×75 Ajax型盲栓连接件
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表 9 压型钢板可拆卸连接件组合梁弯曲试验汇总
Table 9. Summary of bending tests on composite beams with profiled steel decking demountable shear connectors
文献来源 年份 连接件类型 设计要素 跨径/m 加载 混凝土 连接件/mm 钢梁/mm [44] 2014 压型钢板+焊钉螺栓连接件 2.0 正弯矩三点 C20 4行×1列/国标8.8/公称直径20 工字钢S355/UB254×102×28 5.0 正弯矩四点 C20 15行×1列/国标8.8/公称直径20 工字钢S355/UB305×165×46 10.0 正弯矩四点 C20 22行×1列/国标8.8/公称直径20 工字钢S355/UB254×102×28 [46] 2017 压型钢板+焊钉螺栓连接件 5.2 正弯矩四点 C40 26行×1列/ML15/公称直径17 工字钢S355/UB203×203×52 [132] 2023 压型钢板+焊钉螺栓连接件 5.2 正弯矩四点 C40 34行×1列/国标5.8/公称直径17 工字钢Q345/HN350×175×7×11
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表 10 新型可拆卸连接件组合梁弯曲试验汇总
Table 10. Summary of bending tests on composite beams with novel demountable connectors
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表 11 埋入式螺栓连接件抗剪强度
Table 11. Shear strength for embedded bolt connectors
公式编号 文献来源 年份 公式 (1) [151]、[152] 2001、2003 $\begin{gathered} V_{\mathrm{u}}=0.76 A_{\mathrm{s}} f_{\mathrm{u}}+\eta f_{\mathrm{c}} d_{\mathrm{wc}} l_{\mathrm{wc}} \\ \eta=1.5 \end{gathered}$ (2) [153] 2004 $V_{\mathrm{u}}=5.4 A_{\mathrm{s}}\left(f_{\mathrm{c}} E_{\mathrm{c}}\right)^{0.3}$ (3) [154] 2008 $\begin{gathered} V_{\mathrm{u} 1}=0.43 A_{\mathrm{s}} \sqrt{f_{\mathrm{c}} E_{\mathrm{c}}} \\ V_{\mathrm{u} 2}=3 \lambda A_{\mathrm{s}} f_{\mathrm{u}}\left(\frac{E_{\mathrm{c}}}{E_{\mathrm{s}}}\right)^{0.4}\left(\frac{f_{\mathrm{c}}}{f_{\mathrm{u}}}\right)^{0.2} \\ \lambda= \begin{cases}6-h_{\mathrm{s}} /\left(1.05 d_{\mathrm{s}}\right) & \varphi_{\mathrm{s}}<5 \\ 1 & 5 \leqslant \varphi_{\mathrm{s}}<7 \\ h_{\mathrm{s}} / d_{\mathrm{s}}-6 & \varphi_{\mathrm{s}} \geqslant 7\end{cases} \end{gathered}$ (4) [155] 2010 $\begin{gathered} V_{\mathrm{u} 1}=17 A_{\mathrm{s}} f_{\mathrm{c}}^{0.45} E_{\mathrm{c}}^{0.04}, V_{\mathrm{u} 2}=6.2 A_{\mathrm{s}}\left(f_{\mathrm{c}} E_{\mathrm{c}}\right)^{0.2} \\ V_{\mathrm{u} 3}=18 A_{\mathrm{s}} f_{\mathrm{c}}^{0.5} h_{\mathrm{s}}^{0.2}, V_{\mathrm{u} 4}=9 \lambda d_{\mathrm{s}}^{1.4} f_{\mathrm{c}}^{0.5} h_{\mathrm{s}}^{0.6} \end{gathered}$ (5) [156]、[157] 2016 $\begin{gathered} V_{\mathrm{u}}=0.76 A_{\mathrm{s}} f_{\mathrm{u}}+\eta f_{\mathrm{c}} d_{\mathrm{wc}} l_{\mathrm{wc}} \\ \eta=2.5 \end{gathered}$ (6) [24] 2017 $V_{\mathrm{u}}=0.644 A_{\mathrm{sc}} f_{\mathrm{u}}, \frac{h_{\mathrm{s}}}{d_{\mathrm{s}}} \geqslant 1.5$ (7) [158] 2017 $V_{\mathrm{u}}=0.22\left(\frac{h_{\mathrm{s}}}{d_{\mathrm{s}}}\right)^{0.4} A_{\mathrm{sc}} \sqrt{f_{\mathrm{c}} E_{\mathrm{c}}}$ (8) [159] 2018 $V_{\mathrm{u}}=A_{\mathrm{s}} f_{\mathrm{u}}+0.16 \eta f_{\mathrm{c}} d_{\mathrm{wc}}^{2}$ (9) [160] 2019 $V_{\mathrm{u}}=\frac{A_{\mathrm{s}} \sqrt{f_{\mathrm{c}} E_{\mathrm{c}}} h_{\mathrm{s}}}{1.284 t_{\mathrm{p}}+1.573 \sqrt{A_{\mathrm{sr}}}} \leqslant 0.9 A_{\mathrm{s}} f_{\mathrm{u}}$ (10) [161] 2021 $\mathrm{V}_{\mathrm{u}}=0.43 A_{\mathrm{s}} \sqrt{f_{\mathrm{c}} E_{\mathrm{c}}} \leqslant 1.36 A_{\mathrm{s}} f_{\mathrm{u}}^{0.85} f_{\mathrm{c}}^{0.15}$ (11) [20]、[21] 2022 $V_{\mathrm{u}}=0.42 A_{\mathrm{s}} \sqrt{f_{\mathrm{c}} E_{\mathrm{c}}} \leqslant 0.72 A_{\mathrm{s}} f_{\mathrm{u}}$ 2023 $V_{\mathrm{u}}=0.605 \Phi_{\mathrm{s}} A_{\mathrm{s}} \sqrt{f_{\mathrm{c}} E_{\mathrm{c}}} \leqslant 1.25 \Phi_{\mathrm{s}} A_{\mathrm{s}} f_{\mathrm{u}}$ (12) [11] 2023 $V_{\mathrm{u}}=0.0047 d_{\mathrm{s}}^{2} f_{\mathrm{G}}^{14}\left(\frac{h_{\mathrm{s}}}{d_{\mathrm{s}}}\right)^{0.3} f_{\mathrm{u}}^{0.5}<0.7 f_{\mathrm{u}} A_{\mathrm{sc}}$ (13) [97] 2024 $\left(\frac{P_{\text {st－t }}}{P_{\text {tl }}}\right)^{1.75}+\left(\frac{P_{\text {st－s }}}{P_{\mathrm{s}}}\right)^{0.5}=1$ (14) [15]、[16] 2023、2024 $\begin{gathered} V_{\mathrm{u}}=0.6 A_{\mathrm{sc}} f_{\mathrm{u}} \\ V_{\mathrm{u}}=C m_{\mathrm{f}} d_{\mathrm{s}} t_{\mathrm{p}} f_{\mathrm{u}} \end{gathered}$
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表 12 贯穿式螺栓连接件抗剪强度
Table 12. Shear strength for through-bolt connectors
公式编号 文献来源 年份 公式 (15) [30] 2014 $\begin{gathered} V_{\text {total }}=V_{\text {friction }}+V_{\text {dowel }}=\left[k_{\mathrm{s}} \sin \left(\alpha_1\right)+\cos \left(\alpha_1\right)\right] m n T_{\text {total }}+V_{\text {dowel }} \\ \left(\frac{V_{\text {dowel }}}{0.6 n A_{\mathrm{s}} f_{\mathrm{u}}}\right)^2+\left(\frac{n T_{\text {total }}}{n A_{\mathrm{s}} f_{\mathrm{u}}}\right)^2 \leqslant 1 \\ V_{\text {dowel }}=0.6 n A_{\mathrm{s}} f_{\mathrm{u}} \sqrt{1-\left(\frac{n T_{\text {total }}}{n A_{\mathrm{s}} f_{\mathrm{u}}}\right)^2} \end{gathered}$ (16) [28] 2015 $\begin{gathered} V_{\mathrm{u} 1}=0.66 A_{\mathrm{sc}} f_{\mathrm{u}} \\ V_{\mathrm{u} 2}=\mu_{\mathrm{f}} \mathrm{k}_{\mathrm{h}} N_{\mathrm{t}} \end{gathered}$ (17) [32] 2020 $V_{\mathrm{u}}=0.6 \alpha_{2} A_{\mathrm{sc}} f_{\mathrm{u}}$ (18) [36] 2020 $\begin{gathered} V_{\mathrm{u}}=0.5 A_{\mathrm{s}} \sqrt{E_{\mathrm{c}} f_{\mathrm{c}}}\left(\frac{d_{\mathrm{s}}}{d_{\mathrm{c}}}\right)^{0.6} A_{\mathrm{pad}}^{0.05} \times 0.25 \\ V_{\mathrm{u}}=0.5 A_{\mathrm{s}} \sqrt{E_{\mathrm{c}} f_{\mathrm{c}}}\left(\frac{d_{\mathrm{s}}}{d_{\mathrm{c}}}\right)^{0.6} \end{gathered}$ (19) [162] 2020 $V_{\mathrm{u}}=\min \left\{0.5 A_{\mathrm{s}} \sqrt{f_{\mathrm{c}} E_{\mathrm{c}}}, 0.76 A_{\mathrm{s}} f_{\mathrm{u}}\right\}$ (20) [163] 2021 $\begin{gathered} V_{\mathrm{u}}=0.76 A_{\mathrm{sc}} f_{\mathrm{u}} \\ V_{\mathrm{u}}=0.76 k_{\mathrm{c}} A_{\mathrm{sc}} f_{\mathrm{u}} \end{gathered}$
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表 14 新型可拆卸连接件抗剪强度
Table 14. Shear strength for novel demountable connectors
公式编号 文献来源 年份 公式 (23) [67] 2017 $V_{\mathrm{u}}=\frac{\pi d_{\mathrm{s}}^{2} f_{\mathrm{u}}}{4}\left\{\frac{0.8}{\cos \left(\beta_{1}\right)}+0.4\left[\sin \left(\beta_{1}\right)+\mu \cos \left(\beta_{1}\right)\right]\right\}$ (24) [57] 2018 $V_{\mathrm{u}}=0.8 A_{\mathrm{sc}} f_{\mathrm{tk}}$ (25) [64] 2018 $\begin{gathered} V_{\mathrm{u}}=F_{\mathrm{p}, \mathrm{~s}}+F_{\mathrm{p}, \mathrm{w}} \\ V_{\mathrm{u}}=2\left(c f_{\mathrm{t}} A_{\mathrm{s}}+\mu f_{\mathrm{yr}} A_{\mathrm{tr}} \kappa\right) \end{gathered}$ (26) [73] 2021 $V_{\mathrm{u}}=0.7 A_{\mathrm{sc}} f_{\mathrm{u}}$ (27) [63] 2022 $V_{\mathrm{u}}=0.6\left(\frac{34}{d_{\mathrm{s}}}\right)^{\mathrm{a}} f_{\mathrm{u}} A_{\mathrm{s}}$ (28) [78] 2022 $V_{\mathrm{u}}=\min \left\{\alpha_{3} A_{\mathrm{t}} \sqrt{E_{\mathrm{c}} f_{\mathrm{c}}}, \alpha_{4} A_{\mathrm{s}} f_{\mathrm{u}}\right\}$ (29) [79] 2022 $V_{\mathrm{u}}=0.30 A_{\mathrm{t}} \sqrt{E_{\mathrm{c}} f_{\mathrm{c}}} \leqslant 0.84\left(\frac{20}{d_{\mathrm{s}}}\right)^{0.84} A_{\mathrm{s}} f_{\mathrm{u}}$ (30) [76] 2024 $V_{\mathrm{u}} / A_{\mathrm{sc}} f_{\mathrm{u}}=a-b d_{\mathrm{s}}$ (31) [87] 2024 $V_{\mathrm{u}}=\min \left\{0.3 A_{\mathrm{s}} \sqrt{E_{\mathrm{c}} f_{\mathrm{c}}}, 0.5 A_{\mathrm{s}} f_{\mathrm{u}}\right\}$ (32) [88] 2024 $V_{\mathrm{u}}=\mu D_{\mathrm{u}} P_{1} n_{\mathrm{b}} \beta_{2} \gamma$
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表 15 可拆卸连接件刚度计算公式
Table 15. Stiffness calculation formulas for demountable shear connectors
公式编号 文献来源 年份 公式 (33) [57] 2018 $K=\frac{0.5 V_{u}}{S}$ (34) [59] 2018 $E I_{\text {eff }}(x)=\left\{1+\left[\frac{E I_{\infty}(x)}{E I_{0}(x)}-1\right]\left[1+\left(\frac{\alpha(x) L}{\pi}\right)^{2}\right]^{-1}\right\} E I_{\infty}(x)$ 2019 $\text { 弯曲刚度: } K_{\mathrm{b}}=\frac{\Delta F}{\Delta W(x=L / 2)}$
$\text { 剪切刚度: } K_{\mathrm{s}}=\frac{\Delta F}{\Delta u(x=0)}$(35) [91] 2020 $K=\left(\frac{l_{\mathrm{f}}}{12 d}\right)^{x}, x= \begin{cases}0.3 & \frac{l_{\mathrm{f}}}{d} \leqslant 12 \\ 0 & \frac{l_{\mathrm{f}}}{d}>12\end{cases}$ (36) [19] 2023 $K=\tan (\theta)=\frac{V_{G}-V_{C}}{S_{G}-S_{C}}$ (37) [26] 2023 $K=\frac{V_{u}}{S(n)}$ (38) [11] 2023 $K=d_{\mathrm{s}}^{1.12}\left(1.22+0.86 f_{\mathrm{G}}^{9}{ }^{3}\right)\left(\frac{h_{\mathrm{s}}}{d_{\mathrm{s}}}\right)^{-0.083}$ (39) [25]、[126] 2024 $K=\frac{E_{\mathrm{s}} I_{\mathrm{eq}}}{1+\zeta}$
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表 16 可拆卸连接件荷载-滑移曲线模型
Table 16. Load-slip curves models for demountable shear connectors
公式编号 文献来源 年份 公式 (40) [137] 1971 $\frac{V}{V_{\mathrm{u}}}=\left(1-\mathrm{e}^{-0.71 S}\right)^{2 / 5}$ (41) [57] 2018 $V=V_{\mathrm{u}}\left(1-\mathrm{e}^{A S}\right)$ (42) [28] 2014 $V(S)=V_{1}+\left(V_{\mathrm{u}}-V_{1}\right)\left\{1-\exp \left[-0.005 f_{\mathrm{c}}\left(S-S_{1}\right)\right]\right\}^{0.8}$ (43) [162] 2020 $\frac{V}{V_{\mathrm{u}}}=\left(1-\mathrm{e}^{-0.2 S}\right)^{0.4}$ (44) [32] 2020 $\frac{V}{V_{\mathrm{u}}}= \begin{cases}0.137 S & S \leqslant 0.59 \mathrm{~mm} \\ \frac{S-0.38}{2.08+0.87 S} & 0.59 \mathrm{~mm}<S<18.9 \mathrm{~mm}\end{cases}$ (45) [35] 2022 $\text { 国标 8．8／10．9 螺栓 }: V / V_{\mathrm{u}}=\left(1-\mathrm{e}^{-0.47 S}\right)^{0.36}$
$\text { 国标 } 12.9 \text { 螺栓 }: V / V_{\mathrm{u}}=\left(1-\mathrm{e}^{-0.47 S}\right)^{0.5}$(46) [164] 2022 $\begin{gathered} V= \begin{cases}V_{\mathrm{s}} & 0 \leqslant S \leqslant S_{\mathrm{s}} \\ V_{\mathrm{s}}+\left(V_{\mathrm{u}}-V_{\mathrm{s}}\right) \frac{2\left(S-S_{\mathrm{s}}\right)}{S+S_{\mathrm{u}}-2 S_{\mathrm{s}}} & S_{\mathrm{s}}<S \leqslant S_{\mathrm{u}}\end{cases} \\ V_{\mathrm{S}}=2 \mu P_{\mathrm{t} 2} \end{gathered}$ (47) [97] 2024 $\frac{V}{V_{\mathrm{u}}}= \begin{cases}\frac{d_{\mathrm{s}}^{0.6} S}{S_{\mathrm{u}}} & S \leqslant \frac{V_{\mathrm{s}}}{V_{\mathrm{t}} d_{\mathrm{s}}} \\ \left(1.3-\mathrm{e}^{\frac{-1.4 S}{S_{\mathrm{u}}}}\right)^{1.2} & S>\frac{V_{\mathrm{s}}}{V_{\mathrm{t}} d_{\mathrm{s}}}\end{cases}$ $V / V_{\mathrm{u}}=\left(1-\mathrm{e}^{-2.73 S / \mathrm{S}_{\mathrm{u}}}\right)^{0.68}$ $V / V_{\mathrm{u}}=\left(1-\mathrm{e}^{-0.205}\right)^{0.4}$ (48) [76] 2024 $\frac{V}{V_{\mathrm{u}}}=\left(1-\mathrm{e}^{-0.04 S d_{\mathrm{s}}^{1.65}}\right)^{0.0056 d_{\mathrm{s}}^{1.65}}$ (49) [71] 2024 $V_{\mathrm{u}}=V_{1}+\left(V_{\mathrm{u}}-V_{1}\right)\{1-\exp [-0.557(S-\Delta)]\}^{1.004}$
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