Soil that t support foundation subjected to net stresses increases. Net stresses increases depend on

Size: px

Start display at page:

Download "Soil that t support foundation subjected to net stresses increases. Net stresses increases depend on"

Iris Higgins
6 years ago
Views:

1 Stresses in a Soil Mass updated pril 15, 008 Haro Dwito rmono, M.Eng, Ph.D Soil that t support foundation subjected to net stresses increases Net stresses increases depend on oad per unit area to which the foundation is subjected Depth below foundation at which the stress estimation is desired This topic discusses the principles of estimation of vertical stress increase in soil due to various tpes of loading.

2 Simplicit ased on inear theor of elasticit ssumptions: oussines Method J. oussines, 1885 The sstem (loads & soil) in a state of euilibrium ll loads have been applied graduall and no kinetic energ passed on The sstem is conservative and independent of time *) The soil is weiightless, continuous, homogeneous, isotropic, and linearl elastic *) The material constants are known from eperiments and independent of time *) *) cannot be satisfied b the real sstem oussines (1883) Point oad P load - kn r Δp 3 3P Δ p r + 5 π Δ p QI. + + r + Δp Δp P P Q 3 1 I π ( r/ ) + 1 kn 5/ Δ p kn / m kpa

3 ine oad Infinite ength load/unit length - kn/m Δ p π Δ p QI. 3 ( + ) Δp Q I π ( / ) + 1 kn / m Δ p kn / m kpa load/unit area - kn/m Δp β δ Strip oad Finite width, Infinite ength 1 1 tan tan ( /) ( /) + Δ p π ( /4) + ( /4) + Δ p QI. β δ Q I [ β + sin β cos( β + δ) ] π β and δ in radians kn / m 0< β < π π Δ p kn / m < δ < kpa π

4 R load/unit area -kn/m Δ p Circular oad Under circular center 1 1 / + 1 ( R ) 3/ Δ p QI. R Q I 1 1 ( R ) / + 1 3/ Δp kn / m Δ p kn / m kpa I 1.5 Circular oad n location o load / unit area kpa r 8 ; /r 4 7 ; /r 3.5 Figure 8.

5 Under rectangular corner load/unit area -kn/m Δp Rectangular oad Finite width, Finite ength mn m + n + 1 m + n + m + n + m n + 1 m + n + 1 Δ p 4π 1 mn m + n tan m n m n m n given in a chart in Figure. 3.5 Δ p QI. m, n interchangeable Q mn m + n + 1 m + n + 1 m + n + m n + 1 m + n + 1 I 4π 1 mn m + n + 1 tan + m + n m n + 1 N / m Δ p N / m Pa o load / unit area kpa Rectangular oad :/ 0.3 :/ 0.4 I 0.46 Figure 8.1 Figure 3.5

6 Triangular oad Finite width, Finite ength o load / unit area kpa m n given in a chart in Figure. 8.4 Δ p QI. load/unit area -kn/m Q N / m Δ p N / m Pa Δp Figure 8.4 o load of embankment/area kn/m a b Trapeoidal oad Finite width, Infinite ength Δ p QI. β Δp δ h Q ρ. g. h o 1 a+ b b I ( β δ ) δ π a + a o kn / m kpa Δ p kn / m kpa

7 Trapeoid o load / unit area kpa Figure 8.3 Superposition For elastic material superposition is valid /unit length - kn/m /unit length - kn/m /unit length - kn/m C D C D + C D + C D + - 3/4

8 / + 1/ C - + C -

Newmark s Influence Chart dopt a scale such that or Q is eual to depth () Plot th eplan of loaded area on the chart Place the plan in such a wa that point P is located directl above the center of the

. N I influence value pressureon the loaded area (kpa) n number of blocks ssignment 1 vertical structural load F is to be applied above point at the surface of a mass of elastic soil, as shown on

F is applied as a point load at point (Figure a). F is uniforml distributed along a line of length 8m (Figure b) 3. F is uniforml distributed over an area of length 8m and width 0.5m (Figure c) a.

9 Newmark s Influence Chart dopt a scale such that or Q is eual to depth () Plot th eplan of loaded area on the chart Place the plan in such a wa that point P is located directl above the center of the chart Count the number of blocks N Calculate Δp Δ p I.. N I influence value pressureon the loaded area (kpa) n number of blocks ssignment 1 vertical structural load F is to be applied above point at the surface of a mass of elastic soil, as shown on Figure 1. Due to the presence of a soft laer at m depth, an estimate of the vertical stress increase Δp is needed at points and indicated in the Figures. Compute Δp at and for the following cases: 1. F is applied as a point load at point (Figure a). F is uniforml distributed along a line of length 8m (Figure b) 3. F is uniforml distributed over an area of length 8m and width 0.5m (Figure c) a. Treat the area as a strip footing b. Treat the area as a finite rectangular fleible foundation 4. F is uniforml distributed over a circular area centered at so that the contact pressure is the same as in case 3 (Figure d) 5. F is uniforml distributed over a suare area centered at so that the contact pressure is the same as in case 3 and 4 (Figure e) 6. F is uniforml distributed over a trapeoid load of a. Treat the area as a long embankment (Figure f) b. Treat the area as a finite trapeoidal embankment (Figure g) Present our calculation for all cases (8 of them) and summarie our results in a Table ( ) for points and

10 F 100 kn 1 F 100 kn a F 100 kn b 8m m m m m m m c F 100 kn d e F 100 kn F 100 kn 8m 0.5m m m m m m m ssignment highwa embankment as shown in Figure 3a and 3b. ssume the average densit of the material in the embankment is.0 Mg/m 3 Due to the presence of a soft laer at m depth, an estimate of the vertical stress increase Δp is needed at points and indicated in the Figures 3. Compute Δp at and for the following cases: oad is uniforml distributed over an area of length 8m and width 8m (Figure 3a) a. Treat the embankment as a ver long embankment b. Treat the embankment as a finite rectangular and triangular fleible embankment ρ Mg/m 3 ρ Mg/m 3 8m m m 1 1m 8m m 1m 8m 1 m

11 ibliograph Holt, R.D and Kovacs, W.D., n Introduction to Geotechnical Engineering Das,.M., Soil Mechanics Das,.M., dvanced Soil Mechanics Problem Eamples Holt and Kovacs

12 The 3 4 m rectangular footing is loaded uniforml b 117 kpa Reuired : a. Find the vertical stress under the corner of the footing at a depth of m b. Find the vertical stress under the center of the footing at a depth of m Solution: a. 3m, 4m, m; therefore m/ 3/ 1.5, n / 4/ From figure 8.1, I 0.3 Δp o. I kpa Eamples 8.18 a. To compute the stress under the center, it is necessar to divide the 3 4 rectangular footing into 4 sections of 1.5 m in sie. Find the stress under one corner and multipl this value b 4 to take into account the four uadrants of the uniforml loaded area. 1.5m, m, m; then m / 1.5/ 0.75, n / / 1 The corresponding value of I from Fig 8.1 is Δpp o. I kpa Eample m area uniforml loaded with 100 kpa Reuired: a. Find the stress at a depth of 5m under point in Figure 8.19 b. Find the stress at point if the right half of the 5 10m area were loaded with an additional 100 kpa Solution: a. Refer to Figure 8.19 and the numbered points as shown. dd the rectangles in the following 5 1 manner (+ for loaded areas and for unloaded areas) oaded area (867) result in the loaded rectangle we want Find four separate influence values from Figure 8.1 for each rectangle at a depth of 5 5m, then add and subtract the computed stresses Note that it is necessar to add rectangle 584 because it was subtracted twice as part 5 10 of rectangles 164 and 573 Figure 8.19

13 The computation are shown in the following table oaded area (9610) m/ n/ I Δp Total Δp kpa 5 10 b. When rectangle is loaded with 100 kpa and rectangle 9610 is loaded with 00 kpa, repeat part (a) above to obtain the stress under point at 5m depth for the entire rectangle 867 loaded with 100 kpa. Net, a second set of four rectangles would have to be calculated just as for part (a) above but onl rectangle 9610 would be loaded d with +100 kpa; the others would be 100 kpa. The total Δp euals 0.3 kpa from part (a) plus ( ) or kpa Eample 8.0 circular tank 3.91 m in diameter is uniforml loaded with 117 kpa Reuired a. Compute the stress under the center of the tank at a depth of m below the tank b. Compute the stress under the edge of the tank, also a depth of m Solution a. Refer to Figure 8. : m, r 3.91 / 1.95m, 0; then /r / /r 0/ Find I Δp o. I kpa (This compares eactl with Δpp 74 kpa at the center for a 3 4m rectangular area in Eample 8.18, In both cases, the area is 1m ) b. gain refer to Figure 8., For the edge of the circular loaded area: m, r 1.95m, r 1.95m /r / /r 1.0 Find I 0.33 Δp o. I kpa (This compares with with Δp 6 kpa at the corner for a 3 4m uniforml loaded rectangular area in Eample 8.18, In both cases, the area of the loaded area is the same : 1m )

14 Eample 8.1 highwa embankment, as shown below. ssume the average densit of the material in the embankment is.0 Mg/m 3. Compute the vertical stress under the centerline at depth of 3 and 6m Solution : 5m 5m First, calculate the applied surface stress o and the dimensions of the embankment in terms of a and b 1 3m From Figure 8.3 b 5m a 3m 6m 3m Net calulate the vertical stress for 3m a/ 6/3 b/ 5/ m o ρgh.0mg/m 9.81m/s 3m 59 kpa From Figure 8.3, I 0.49 Δ p oi 59 kpa kp For one half of the embankment, or 58 kpa for the entire embankment. Thus at this shallow depth, Δp is almost the same as the contact stress. Finall, calculate the vertical stress for 6m a/ 6/6 1 b/ 5/ From Figure 8.3, I 0.44 Δ p I 59 kpa kp o Eample 8. uniform load of 50 kpa is applied to the loaded area shown in Figure Compute the stress at depth of 80m below the ground surface due to the loaded area under point 40 Uniform oad 50 kpa Depth Q

15 epth 80 De 0 Depth 80 Draw the loaded area such that the length of the line Q is scaled to 80m. For eample, the distance in Figure above is 1.5 times the distance Q, Q 80m and 10m Net, place point, th epoint where the stress is reuired, over the center of the influence chart The number of blocks (and partial blocks) are counted under the loaded area. In this case, abouth 160 blocks are found. The vertical stress at 80m is then indicated b Δ p I no of blocks o 50kPa blocks 40 kpa To compute the stress at other depths, the process is repeated b making aother drawings for the different depths, changing the scale each time to correspond to the distance Q on the influence chart ibliograph Holt, R.D and Kovacs, W.D., n Introduction to Geotechnical Engineering Das,.M., Soil Mechanics Das,.M., dvanced Soil Mechanics

Subsurface Stresses (Stresses in soils)

Subsurface Stresses (Stresses in soils) ENB371: Geotechnical Engineering Chaminda Gallage Contents Introduction: Subsurface stresses caused by surface loading Subsurface stresses caused by the soil mass