Experimental study on concrete beams without web reinforcement based on fractal theory
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摘要: 基于裂縫的發展及分布形態，探究無腹筋混凝土梁在不同剪跨比和縱筋配筋率作用下的剪切性能，采用剪跨比分別為1.5、2、2.5和縱筋配筋率分別為1.28%、1.62%、1.99%的9組無腹筋混凝土梁進行四點加載受剪試驗，通過應用分形幾何理論對試驗梁表面的裂縫進行分析，使用盒計數法計算得到分級荷載及極限荷載作用下梁表面裂縫的分形維數，探討了梁表面分形維數與極限荷載、分級荷載及跨中撓度之間的關系。結果表明：剪跨比與極限荷載及開裂荷載成反比，而縱筋配筋率與極限荷載成正比，但其對于開裂荷載的影響較小。無腹筋混凝土梁不論在分級加載作用下還是極限荷載作用下都具備明顯的分形特征，在分級荷載作用下的分形維數在0.964～1.449，在極限荷載作用下的分形維數在1.33附近。分級荷載、跨中撓度與分形維數之間呈現較好的對數關系，分級荷載與分形維數的變化曲線受剪跨比及梁縱筋配筋率的影響具有一定的規律性，而跨中撓度受剪跨比的影響較小，在縱筋配筋率作用下，其曲線的曲率呈現出先增大后減小的趨勢，但極限荷載與分形維數之間的關系具有一定的差異性，極限荷載會隨著剪跨比的增大呈現出先增大后減小的趨勢，隨著縱筋配筋率的增大呈現出的差異性較大。Abstract: Based on the development and distribution of cracks, we explored the shear performance of concrete beams without web reinforcement under different shear span ratios and longitudinal reinforcement ratios. Nine groups of concrete beams without web reinforcement with shear-span ratios of 1.5, 2, 2.5 and longitudinal reinforcement ratios of 1.28%, 1.62%, and 1.99% were used for four-point loading shear tests. The cracks on the surface of the test beam were analyzed by applying fractal geometry theory, and the box counting method was used to calculate the fractal dimension of the cracks on the surface of the beam under the effect of the graded load and the ultimate load. The relationship among the fractal dimension of the beam surface, the ultimate load, the graded load and the span was discussed. The results show that the shear-span ratio is inversely proportional to the ultimate load and cracking load, while the longitudinal reinforcement ratio is directly proportional to the ultimate load and exhibit a small influence on the cracking load. Concrete beams without web reinforcement have obvious fractal characteristics under the effect of graded loading or ultimate load. The fractal dimension under the effect of graded load is 0.964–1.449, and the fractal dimension under the effect of ultimate load is around 1.33. The graded load, mid-span deflection and fractal dimension show a good logarithmic relationship. The change curve of graded load and fractal dimension is affected by the shear-span ratio and the beam longitudinal reinforcement ratio. The intermediate deflection is less affected by the shear-span ratio. Under the effect of the longitudinal reinforcement ratio, the curvature of the curve shows a trend of first increasing and then decreasing, but the relationship between the ultimate load and the fractal dimension has certain differences. The ultimate load first increases and then decreases with the increase of the shear span ratio, and the difference is greater with the increase of the longitudinal reinforcement ratio.
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圖 1 無腹筋梁尺寸及配筋圖（單位: mm）
Figure 1. Dimensions and reinforcement drawing of girder without rib (Unit: mm)
圖 2 加載裝置布置圖。（a）λ = 1.5；（b）λ = 2；（c）λ = 2.5；（d）現場布置圖
Figure 2. Load device layout: (a) λ = 1.5; (b) λ = 2; (c) λ = 2.5; (d) site layout
圖 3 試驗梁裂縫分布圖。（a）WL-1；（b）WL-2；（c）WL-3；（d）WL-4；（e）WL-5；（f）WL-6；（g）WL-7；（h）WL-8；（i）WL-9
Figure 3. Crack distribution of test beam: (a) WL-1; (b) WL-2; (c) WL-3; (d) WL-4; (e) WL-5; (f) WL-6; (g) WL-7; (h) WL-8; (i) WL-9
圖 4 相同剪跨比、不同縱筋配筋率作用下的荷載與撓度間關系。（a） λ = 1.5；（b）λ = 2；（c）λ = 2.5
Figure 4. Relationship between load and deflection under the same shear-span ratio and different longitudinal reinforcement ratios: (a) λ = 1.5; (b) λ = 2; (c) λ = 2.5
圖 5 相同縱筋配筋率、不同剪跨比作用下荷載與撓度間的關系。（a）ρ = 1.28%；（b）ρ = 1.62%；（c）ρ = 1.99%
Figure 5. Relationship between load and deflection under the same longitudinal reinforcement ratio and different shear span ratios: (a) ρ = 1.28%; (b) ρ = 1.62%; (c) ρ = 1.99%
圖 7 不同等級荷載下梁表面的lnN(L)–ln(1/L)圖。（a）WL-1；（b）WL-2；（c）WL-3；（d）WL-4；（e）WL-5；（f）WL-6；（g）WL-7；（h）WL-8；（i）WL-9
Figure 7. lnN(L)–ln(1/L) diagram of beam surface under different grades of load: (a) WL-1; (b) WL-2; (c) WL-3; (d) WL-4; (e) WL-5; (f) WL-6; (g) WL-7; (h) WL-8; (i) WL-9
圖 8 極限荷載下梁表面的lnN(L)–ln(1/L)圖
Figure 8. lnN(L)–ln(1/L) diagram of the beam surface under ultimate load
圖 10 相同剪跨比、不同縱筋配筋率作用下的極限荷載與分形維數間的關系。（a）λ = 1.5；（b）λ = 2；（c）λ = 2.5
Figure 10. Relationship between ultimate load and fractal dimension under the same shear span ratio and different longitudinal reinforcement ratios: (a) λ = 1.5, (b) λ = 2, (c) λ = 2.5
圖 11 相同縱筋配筋率、不同剪跨比作用下的極限荷載與分形維數間的關系。（a）ρ = 1.28%；（b）ρ = 1.62%；（c）ρ = 1.99%
Figure 11. Relationship between ultimate load and fractal dimension under the same longitudinal reinforcement ratio and different shear span ratios: (a) ρ = 1.28%; (b) ρ = 1.62%; (c) ρ = 1.99%
圖 12 相同剪跨比、不同縱筋配筋率作用下的分級荷載與分形維數間關系。（a）λ = 1.5；（b）λ = 2；（c）λ = 2.5
Figure 12. Relationship between the graded load and the fractal dimension under the same shear span ratio and different longitudinal reinforcement ratios: (a) λ = 1.5; (b) λ = 2; (c) λ = 2.5
圖 13 相同縱筋配筋率、不同剪跨比作用下的分級荷載與分形維數間關系。（a）ρ = 1.28%；（b）ρ = 1.62%；（c）ρ = 1.99%
Figure 13. Relationship between the graded load and the fractal dimension under the same longitudinal reinforcement ratio and different shear span ratios: (a) ρ = 1.28%; (b) ρ = 1.62%; (c) ρ = 1.99%
圖 14 相同剪跨比、不同縱筋配筋率作用下的跨中撓度與分形維數間的關系。（a）λ = 1.5；（b）λ = 2；（c）λ = 2.5
Figure 14. Relationship between mid-span deflection and fractal dimension under the same shear-span ratio and different longitudinal reinforcement ratios: (a) λ = 1.5; (b) λ = 2; (c) λ = 2.5
圖 15 相同縱筋配筋率、不同剪跨比作用下的跨中撓度與分形維數間關系。（a）ρ = 1.28%；（b）ρ = 1.62%；（c）ρ = 1.99%
Figure 15. Relationship between mid-span deflection and fractal dimension under the same longitudinal reinforcement ratio and different shear-span ratios: (a) ρ = 1.28%; (b) ρ = 1.62%; (c) ρ = 1.99%
表 1 水泥的化學成分(質量分數)
Table 1. Chemical composition of cement
% SiO2 Al2O3 Fe2O3 CaO MgO SO3 K2O Na2O Li2O 21.22 5.05 3.26 60.24 0.97 2.67 0.50 0.73 — 
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表 2 粗骨料的實驗性能
Table 2. Properties of coarse aggregate
Micron content/% Water absorption rate/% Needle-like content/% Ruggedness/
%Apparent density/(kg·m–3) 0.3 0.5 5 1 2640
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表 3 試件參數信息
Table 3. Parameter information of test pieces
Numbering Size/mm Compressive strength/MPa Reinforcement diameter/mm Longitudinal strength Shear span ratio Reinforcement ratio/% WL-1 1800×150×250 41.031 16 HRB400 1.5 1.28 WL-2 1800×150×250 44.103 16 HRB400 2 1.28 WL-3 1800×150×250 42.772 16 HRB400 2.5 1.28 WL-4 1800×150×250 38.467 18 HRB400 1.5 1.62 WL-5 1800×150×250 37.003 18 HRB400 2 1.62 WL-6 1800×150×250 36.832 18 HRB400 2.5 1.62 WL-7 1800×150×250 38.607 20 HRB400 1.5 1.99 WL-8 1800×150×250 38.471 20 HRB400 2 1.99 WL-9 1800×150×250 46.219 20 HRB400 2.5 1.99
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表 4 不同荷載作用下梁表面分形維數
Table 4. Fractal dimension of beam surface under different loads
Load/kN Fractal dimension WL-1 WL-2 WL-3 WL-4 WL-5 WL-6 WL-7 WL-8 WL-9 20 — — 0.964 — — 0.951 — — — 40 1.018 1.044 1.040 0.991 1.042 1.082 0.991 1.058 0.955 60 1.059 1.1 1.275 1.044 1.075 1.221 0.999 1.072 1.122 80 1.093 1.21 1.336 1.133 1.215 1.330 1.054 1.170 1.161 100 1.164 1.32 — 1.203 1.283 1.449 1.157 1.248 1.321 120 1.267 — — 1.208 1.340 — 1.188 1.295 — 140 1.304 — — 1.218 1.357 — 1.193 1.354 — 160 1.302 — — 1.276 — — 1.207 — — 180 1.335 — — 1.356 — — 1.255 — — 190 — — — — — — 1.333 — —
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表 5 分級荷載與分形維數關系的k、m值
Table 5. k, m values of the relationship between the graded load and the fractal dimension
Parameters WL-1 WL-2 WL-3 WL-4 WL-5 WL-6 WL-7 WL-8 WL-9 k 0.156 0.149 0.113 0.150 0.141 0.082 0.140 0.120 0.148 m 8.984 –6.25 6.25 8.827 –1.481 27.037 8.693 4.824 –20.653 R2 0.926 0.896 0.892 0.943 0.942 0.890 0.898 0.917 0.921
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表 6 跨中撓度與分形維數關系的n、v值
Table 6. n, v values of the relationship between the mid-span deflection and the fractal dimension
Parameters WL-1 WL-2 WL-3 WL-4 WL-5 WL-6 WL-7 WL-8 WL-9 n 0.129 0.108 0.116 0.123 0.097 0.056 0.126 0.102 0.093 v 83.047 82.727 73.281 66.049 68.457 75 51.307 52.613 48.898 R2 0.937 0.939 0.966 0.950 0.937 0.922 0.928 0.904 0.966
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