Terramechanics 2. Soil bearing parameters Bulldozing Tandem wheels MARYLAND U N I V E R S I T Y O F

Transcription

1 Soil bearing parameters Bulldozing Tandem wheels 1 01 David L. Akin - All rights reserved

2 Terzaghi Soil Bearing Capacity Factors N q = exp 3π φ tan φ cos π 4 + φ exp 3π N c = cot φ φ tan φ 1 cos π 4 + φ = cot φ (N q 1) N γ = 1 Kpγ cos φ 1 tan φ K pγ = 8φ 4φ +3.8 tan π 3 + φ

3 K c Modulus of density of soil deformation 3 K c =(N c tan φ) cos φ K γ Modulus of cohesion of soil deformation Nγ K γ = tan φ +1 cos φ α Angle of approach of wheel to soil α = cos 1 1 z D N γ Weight density of soil m 3

4 Soil Bearing Limit Safe weight on the soil W s = A cn c + γzn q + 1 γbn γ c Soil cohesion (Pa) b Wheel width (m) 4

5 Bulldozing Resistance General case: R b = b sin (α + φ) sinα cos φ zckc + γz K γ + π3 oγ( π φ) 3π + cπ o π + c o tan π 4 + φ For tracked vehicles, only the first term applies: R b = o = z tan π 4 φ b sin (α + φ) sinα cos φ 5 zckc + γz K γ

6 Tandem Wheels W W 1 D D 1 θ 0 θ 1 θ 0 z 1 z 0 z z 0 = z 1 + z Assume n = 1 = P = k z P 1 = k z 1 P = k z 1 + z 6

7 Soil Weight Bearing Analysis In general, W = θ0 0 df cos θ = θ0 0 Pb dscos θ W = θ0 0 bk z cos θds ds = rdθ z = r(cos θ cos θ 0 ) W = θ0 0 bkr r(cos θ cos θ 0 ) cos θdθ 7

8 Generic Wheel Soil Suspension Assuming small sinkage, z small, θ small, cos θ 1 cos θ 1 θ cos θdθ dθ + (higher order terms) W = bkr3/ θ0 0 θ 0 θ dθ W = bkr3/ 1 θ 0 sin 1 θθ0 + θ θ 0 θ θ0 0 8

9 Weight on the Front Wheel W = πbkr3/ 4 θ 0 Front wheel: z 1 = r 1 r 1 cos θ 0 z 1 = r 1 r 1 1 θ 0 + = θ 0 = z 1 r 1 W 1 = πbkz 1 r1 9

10 Weight on Back Wheel Change to limits of integration: 0 θ 0, r r W = bkr3/ θ0 θ 0 θ = θ θ 0 θ dθ θ θ 0 + θ0 0 θ 0 θ dθ = θ 0 θ θ 3 1 θ 0 = θ 0 θ θ 1 θ 0 = θ 0 θ 1 10

11 Weight on Back Wheel W = bkr3/ θ 0 θ 1 z 0 = r θ 0 θ 0 = z 0 r θ 1 = z θ = 1 r z r W = bk r z0 z 11

12 Track Depth of Tandem Wheels Much algebra then ensues... Front: z 1 = W 1 πbk r 1 W Back: z = bk 1 r z0 z 0 = z 1 + z = W 1 πbk r 1 + W (bk) r 1 z 0 z 0 W 1 πbk r 1 z 0 + W (bk) r =0 1

13 Rolling Resistance of Tandem Wheels Solve the quadratic equation to get z 0 = 1 W1 bk π + r 1 W 1 π r 1 + W r This was all done for n = 1 = R = 3 bkz3/ 0 R = 3 R = 3 1 W1 bk π + r 1 1 W 1 bk π + D 1 W 1 π r 1 + W r 4W 1 π D 1 + W D 3/ 3/ 13

14 Nondimensional Forms Total wheel load W = W 1 + W Wheel weight ratio a W 1 W For W 1 = W = W = a =1 R = 3 W 1 = a 1+a W W = 1 1+a W 1 W 3/ a (a + 1) 3/ bk π + D 1 4a π D D 3/ Define wheel diameter ratio ρ D 1 D 14

15 Nondimensional Forms R = 3 1 W 3/ (a + 1) 3/ bk D 3/4 a π ρ a π ρ 3/ Let ξ 3 1 (a + 1) 3/ a π ρ a π ρ 3/ R = ξ bk W 3/ D 3/4 15

16 Simple Example Case Consider ρ =1 (D 1 = D = D) a =1 W 1 = W = W For tandem wheels, R = bk W 3/ D 3/4 For single wheel (n=1/), R = bk W 3/ D 3/4 Tandem wheels reduce rolling resistance by 34% 16

17 Dual Wheels Equivalent to single wheel case twice as wide b = b R = bk W 3/ D 3/4 R dual = bk W 3/ D 3/4 Dual wheel rolling resistance 9% less than single, 7% higher than tandem 17

18 ATHLETE in Partial Gravity Suspension UNIVERSITY OF 18