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54 5 4.5 4 3.5 B F l ow 3 L L -4 k sh 2.5 k = 1 . 15 + 7 e B sh 2 1.5 1 0.5 0 0 1 2 3 4 5 6 7 8 9 10 11 12 L /B Figure 6.21. Relationship between ksh (= max/max(circle)) and L/B for deep water (H/B > 2). 6.13 MAXIMUM SHEAR STRESS EQUATION FOR COMPLEX PIER SCOUR B In the previous sections, individual effects on the maximum shear stress are studied by numerical simulations. A series of Flow L figures and equations are given to quantify the corresponding correction factors. However, bridge piers are likely to exhibit a combination of these effects and recommendations are needed to combine these effects in the calculations. It is rec- Figure 6.22. Definition of the angle of attack problem. ommended that the correction factors be multiplied in order to represent the combined effect. This common approach implies that the effects are independent and has been used in many proposed for the correction factor ka giving the influence of instances. the attack angle on the maximum bed shear stress is The proposed equation for calculating the maximum shear stress for a complex pier before the scour process starts is max 0.57 ka = = 1 + 1.5 90 (6.5) max (0 deg) max = kw ksp ksh ka 0.094 V 2 - 1 1 (6.6) log Re 10 3.5 3 2.5 3.5 2 Flow = 1.05 3 = 1. 36 2.5 0.93 1.5 1.55 1.29 1.65 88 2 2.00 1 1. Flow 1.5 0.5 0.24 1 Y/B 0 Pier 0.5 0.75 Y/B -0.5 0 -0.5 0.75 -1 0.93 2.26 -1.5 -1 0.24 2.11 1.88 -1.5 -2 2.90 1.55 -2 0.93 -2.5 1.36 2.36 1.05 -2.5 1.11 1.83 -3 1.47 0.93 9 -3 1.2 -3.5 -3.5 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 6 X/B X/B Figure 6.23. Bed shear stress (N/m2) contours for an Figure 6.24. Bed shear stress (N/m2) contours for an attack angle equal to 15 degrees. attack angle equal to 30 degrees.

OCR for page 54
55 where 3.5 3 2.03 B is the pier width (m); 2.5 = 1.61 2.52 H is the water depth (m); 2 Flow 1.33 1.08 V is the upstream velocity (m/s); 1.5 is the density of water (kg/m3); 1 is the attack angle (in degrees); 0.5 0.33 0.33 VB Re is the Reynolds Number, defined as R e = ; Y/B 0 v Pier v is the kinematic viscosity (m2/s); -0.5 0.33 1.33 kw is the correction factor for the effect of water depth, 0.33 2.52 -1 max defined as kw = and the equation is -1.5 max ( deep) -2 2.03 3.27 4H -2.5 2.5 3.00 - 8 2 kw = 1 + 16e 1.0 1.61 1.33 B ; -3 2.32 ksp is the correction factor for the effect of pier spacing, -3.5 -2 -1 0 1 2 3 4 5 6 max X/B defined as ksp = and the equation is max (single) Figure 6.25. Bed shear stress (N/m2) contours for an S -1.1 attack angle equal to 45 degrees. ksp = 1 + 5e ; B ksh is the correction factor for the effect of pier shape, max defined as ksh = and the equation is max (circle) max L defined as ka = and the equation is -4 max (0 deg) ksh = 1.15 + 7e ; B ksh = 1 for circular shape; and 0.57 ka = 1 + 1.5 90 ka is the correction factor for the effect of attack angle, 3 2.5 2 k 1.5 0 .5 7 F l ow k a = 1 +1 . 5 B 1 90 L 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 /90 Figure 6.26. Relationship between ka (= max/max(0 degree)) and for deep water (H/B > 2).