Sfresses in a Soil Mass

9,1

Construct ion of a foundat ion causes changes in thc st ress, usunl ly a net incrcase. The

net s t ress increase in the soi l c lcpcnds on thc load per uni t arca to which the founda-

t ion is subjected, thc c lepth below the foundat ion at which thc st ress cst imat ion is de-

s i red, and other larctors. l t is nccessary to cst imate the nct increase of 'ver t ica l s t ress

in soi l that occurs as a resul t o f the construct ion of a foundat ion so that set t lemcnt

can be calculated. Thc set t lemcnt calcul i t t ion procedut 'c is d iscussed in more deta i l

in Chapter 10. This chnptcr c l iscusscs thc pr i r rc ip les o1 'es l imat ion o l 'vcr t ica l s t ress

incrcase in soi l caused by var ious typcs of loading, based i tn thc theory of c last ic i ty .

Al though natura l so i l dcposi ts , in most cases, arc not l 'u l ly c last ic , isot ropic . or ho-

mogencous mater ia ls , ca lculat ions for est inrat ing incrcase s in vcr t ica l s t ress y ic ld

fairly good results l 'or practical work.

Normal and Shear Sfresses on a Plane

Students in a soi l mechanics course arc fami l iar wi th the fundamental pr inc ip les of

thc mechanics of del'ormable solids. This section is a bricf rcview of the basic con-

ccpts of normal ancl shcar stresses on a plane that can be found in any course on the

mcchanics o l ' mater ia ls .F igure 9. 1a shows a two-dimensional so i l e lcment that is being subjected to nor-

mal and shear stresses (o,, ) rr,). To dctermine thc normal stross and the shear stress

on a plane EF lhat makes an angle 0 with the plane z1B, we need to consider the tiee

body diagram of EFB shown in Figure 9.1b. Let o,, and r,, be the normal stress and

thc shear stress, respectivcly, on the plane Efi From geometry, we know that

and

EB : EF c<ts 0

F B : F , F s i n 0

Summing the components of forces that act on the elemcnt in the direction of ly' and

4 we have

(e 1)

(e.2)

224

o,,(EF) : c,@F) sin2 0 + o,(EF) cos2 0 + 2r, ,(EF) sin d cos g

9.1 Normal and Shear Stresses on a plane 225

t +It"

T r t

B

( a )

or .

( b )

F i g u r e 9 ' 1 ( a ) A s o i l c l e m c n t w i t h n o r m a l a n d s h c a r s t r c s s c s a c l i n g o n i t ; ( b ) f r e c b o d vdiagram of EFB as shown in (a)

( r , : ( r \ s in2 0 + ( r l . cos2 0 I 2r ,us in { i cos 0

o, =,? . rycos 20 * r,rsin2o

Again,

r , , (EF ) - - o , (E I t ) s i n 0 cos 0 + c , (EF) s i n 0 cos 0

r , , (EF ) cos r o + r , , ( 27 ; s i n2 g

r , - ( F ] . s i n 0 c o s 0 - c , s i n 0 c o s 0 r , , , ( c o s l O - s i n 2 d )

, , : T s i n 2 o - r , r c o s 2 o (e.4)

From Eq. (9.4), we can see that we can choose thc value of g in such a wav thatr,, wil l be equal to zero. Substituting r, : 0, we get

/ t

ran20 : - ' .Yr '

(f| c_,

(e.3)

(e.s)

226 Chapter 9 Stresses in a Soil Mass

For given values of r.,,,, rr.., and o.", Eq. (9.5) will give two values of 0 that are 90" apart.

This means that there are two planes that are at right angles to each other on which

the shear stress is zero. Such planes are called principal planes. The normal stresses

that act on the principal planes are referred lo as principal stresses. The values of

principal stresses can be found by substituting Eq. (9.5) into Eq. (9.3), which yields

Major principal stress :

o, * tf,c n : a l :

Z +

Minor principal stress:

cr, * u,o n : a 3 :

Z *

(e.6)

(:e.7)

The normal stress and shcar stress that act on any plane can also be determined

by plotting a Mohr'.s circle, as shown in Figurc 9.2. The following sign conventions

are uscd in Mohrs circles: compressive normal stresses are taken as positive, and

shcar stresscs are considcrcd positive if they act on opposite faces of the element in

such a way that they tend to produce a counterclttckwise rotation.For plane AD <t[ thc scli l element shown in Figurc 9. 1a, normal stress equals

*rr, arnd shcar stress equals *r.,,. For plane AB, normal stress equals *rr., and shear

stress equals -r.,,,.

The points R and M in Figure 9.2 rcpresent the strcss conditions on planes,4D

and AB, respcctively. O is the point of intcrsection of thc normal stress axis with the

line RM.'I 'he circle M NQRS drawn with O as the center and OR as the radius is the

Shear stress, T

9'l " ' {(O'i " ') ' ' . :

'";l/{K o. ,1'f" x'\ \

\ (o r ' -T r rJ ,

\ _-,'

O 1

M

Figure 9.2 Principles of thc Mohr s circle

Normal stress. o

9.7 Normal and Shear Sfresses on a plane 227

Mohr's circle for the stress conditions considered. The radius of the Mohr,s circle isequal to

(e.8)

(e.e)

The stress on plane E,F canbe determined by moving an angle 20 (which is twice theangle that the plane EFmakes in a counterclockwise clirection with plane AB in Fig-ure 9.1a) in a counterclockwise direction from point M along the circumfcrence ofthe Mohr's circle to reach point Q.The abscissa and ordinate of point e, respectively,give the normal stress o,, and the shear stress r,, on plane 6[

Because the ordinates (that is, the shear stresses) of points ly' ancl .! are zero,they represent the stresses on the principal planes. The abscissa of point ly' is equalto a, [Eq. (9.6) ] , and the abscissa for point S is o j [Eq. (O.Z) ] .

As a specia l casc, i f the p lancs AB and AD were major and min.r pr inc ipalplanes, thc normal stress and the shear stress on plane EFcould bc found by substi-tu t ing r , , : 0 . Equat ions (9.3) and (9.4) show that ( r \ . - ( r t and r r , - o . , (F igure 9.3a) .Thus .

u t + u \ ( r t t r tU tt

' - - t - - COS l t l" 2 2

(Jt * ( f t

7 , , : : ^ - s i n 2 0z

The Mohr's circle for such stress conclit ions is shown in Figure 9.3b. The abscissaand the ordinate of point Q give the normal stress and the shear strcss, respcctivclv.on the plane EF-.

Figure 9.3 (a) Soil element with,4B and AD as major ancl minor principal planes;(b) Mohr's circle for soil element shown in (a)

O 1

J

IO 1

(a)

Chapter 9 Sfresses in a Soil Mass

Example 9.1

A soil element is shown in Figure 9.4. The magnitudes of stresses are or :

120 kN/ m2,r - 40 kN/m2, dr : 300 kN/m2, and 0 : 20'' Determine

a. Magnitudes of the principal stressesb. Normal and shear stresses on plane AB. Use Eqs. (9.3), (9.4), (9.6),

and (9.7).

+,iI

o \ +",

300 + 120

,+o.l

Figure 9.4 Soil element with stresses acting on il

Solutiona. From Eqs. (9.6) and (9.7),

, r . l o, * o '" \ * +( *

o r ) t

',/|ry]'*1-on;'or : 308'5 kNimz

o: : 111.5 kN/m2

b. From Eq. (9..1).

t t , , - o , 0 , - o ,' ' '

, o 'cos

2o -t r s in2oo r : 2

- r - 2

300 + r20 300 - 120: ' " " - - - "

* - cos (z x 20) + ( *40) s in (2 x 20)2 2

:252.23 kN/m2

l o , - o , 1 t

L 2 l

9.2 Sfress Caused by a point Load

From Eq. (7.4),

- - c ) - o ,rn: =- sin 2d - r cos2o

: !9q-l?9 sin (2 x z0) * (_40)cos (2 x 20)

88.49 kN/m2

Sfress Caused by a point Load

Boussinesq (1it iJ3) solvcd the problem of strsc neous, erast ic, an d i sot ropi c m edi um ", ;".::l,l',lff TI,; fi}"::ilffi :;T:;surfacc of an infinitery large trorr-rpu... ,+..oraing to nrgu.. r.s,-B.rrrri,r"rq..s s.ru_tron tor normal stresses at a point caused by the point load p is

Aa,:+{*-(, -r.[ffi.;Arr, : :,{tt:- (r - ,rlr,.a:.i

229

, f \

. v - z l l' t t r . l J

( e . l o )

, - r - ; l l- t . , r , l l ( e . l l )

and

A o . :3P z32 n L s

3P z32n (r2 * fsstz (e . t2)

w h e r c r : V ; r + v ,

L : f ? + y , . . : : V l + , :p : poisson.s rat io

Figure g.S

Stresses in anelastic mediumcaused by apoint load

P

J

230 Chapter 9 Sfresses in a Soil Mass

Note that Eqs. (9.10) and (9.11), which are the expressions for horizontal nor-

mal stresses, depend on the Poissonk ratio of the medium. However, the relationship

for the vertical normal stress, Ao. , as given by Eq. (9.1 2), is independent of Poisson'.s

ratio. The relationship for Ao- can be rewritten as

Lo ,p ( z

r l J

z . ' l 2 nI P .? : - t tI z ' '

(e . l -3 )l ( r tz)z + 11" '

where

3 l1 , - -

- - - ( 9 . 1 - l )' 2 n l ( r t z ) : - I l t '

The variation of 11 lor various values of r/z is given in Tablc 9' 1.

Table 9.1 Variation ol' 11 for Various Values of r/z IEq. (9. l ' l)l

l1rlz rlz

t )

0.020.040.060.0u0 . I 00 . 120 . 1 40 . 1 60 . I l J0.200.220.240.260.2[r0.300.320.340.360.3t10.4t)0.450..500.550.600.6s0.70

( \ .17750.41100.47650.47230.46990.46510.460'70.454u0.44820.44090.43290.42420 .41510.40-500.39540.3r3490.37420.36320.35210.34080.32940.30110.27330.24660.22140.19780.1762

0.7-s0.u00.13-50.900.95l . (x)| .2( l1 .401 .60l . u02.002.202.402.602.803.(X)3.203.403.603.804.004.204.401.604.805.00

0.156-50.13tr60 . t2260.1 0830.09-560.0u440.0-5I30 .03 I70.02000.012r)0.(x)tt50.(x)-5tt0.(n400.(x)290.00210.001-50 .001 r0.0(x)8-50.000660.000-510.000400.000320.000260.000210.000170.00014

r(m)

rIII

IIIIII

i

', g,g

9.3 Vertical Stress Caused bv a Line Load 231

Example 9.2

consider a point load P: 5 kN (Fig. 9.5). Calculate the vertical stress increase( A o . ) a t z : 0 , 2 m . 4 m , 6 m , 1 0 m . a n d 2 0 m . G i v e n x = 3 m a n d v : 4 m .Solution

, * t /FTT = \/7 + 4:5mThe following table can now be prepared:

o',: ($)',lr {kN/m2)

00.00340.04240.12950.27330.4103

z(ml

r2

0 o o) ) <

4 1.256 0.83

10 0.520 0.25

00.00430.01330.01800.01370.0051

Vertical Sfress Caused by a Line Load

Figure 9.6 shows a flexible l ine load of infinite length that has an intensity r7lunit lengthon the surface of a semi-infinite soil mass. The vertical stress increasc, Ao., inside lhesoil mass can be determined by using the principles of the theory of elasticity, or

A o , : , ? ' 1 ' r =1r\x- + z'r(e. r.s.1

Figure 9.6Line load over the surface of asemi-infinite soil mass

232 Chapter 9 Sfresses in a Soil Mass

Table 9.2 Variation ot L,o-l(qlz) with 'vlr [nq. (9'16)]

xlz AcJlqlzl xlz Ao,l(qlzl

00 . 10.20.30.40.-50.60.70.tJ0.91 . 0l . l1 . 2

0.6370.6240.5890..5360.4730.4010.3440.2u70.2370 .1 940 .1 -590 .1 300 . I07

I . - l

1 . 4t . 51 . 61 . 71 . 81 . 92.02.2a ,,1

2 .6

2.tt3.0

0.0utr0.0730.0600.0500.0420.0350.0300.0250.0I 90 .0140.01 I0.00tt0.(x)6

This equation can be rcwrittcn as2q

- \ r r ' - ' . , . ) , )

7 1 7 . 1 \ x t 7 . ) ' l l -

Ao- 2

Q1 lz . ) n l ( x l z )2 + t ) )(e .1 6)

Notc that Eq. (9.16) is in a r " rorrc l imcnsional form. Using th is cquat ion. wc can calcu-

latc thc varialion r-tt L,c-l(qlz.) with r/:. This is given in Table 9.2. The value of Ao"

calculatcd by us ing Eq. (9.16) is thc addi t i r )nals t rcss on soi lcaused by thc l inc load.

The valuc gf Arr- docs not includc thc <lverhurden prcssure ol'the soil above point.4.

Example 9.3

Figure 9.7a shows two line loacls and a point load acting at the ground surface.

Determine the increase in vertical stress at point,4, which is located at a depth

o f 1 .5 m.

SolutionReferring to Figures 9.7b through 9'7d, we find that

Ao. : Ao.11y * Ao121 * Aori3;

2qrz' 2qrz' 3P z3, J x r

? (.ri + z- y '

rr(xl + z")t 2tr (r2 + z')t't ,tr t l

Eq. (e.1.s) Eq. (e.1s) Eq. (e.12)

(1 s) '(2X1sX1.s)3 (2X1oX1.s)3 , (3X30)' t ( 2 f * ( l 5 f l ' * u l ( 4 f + ( l 5 f l ' ? - Q 1 r ,

0.825 + 0.065 + 0.012 = 0.902 kN/m2

{l(3)'+ (4)'l + (1.s)'i"'

2339.4 VerticalSfress Caused by a Strip Load

.,..1:.t.;., .., .

( d )

Figure 9.7 Two line loads and a point load acting at the ground surface

{ = Load per unit area

234 Chapter 9 Sfresses in a Soil Mass

Figure 9.8Vertical stress caused by aflexible strip load

9,4 Vertical Sfress Caused by a Strip Load(Finite Width and Infinite Length)The fundamental equation for the vertical stress increase at a point in a soil massas the rcsult of a l ine load (Section 9.3) can be uscd to determine the vertical stress at

a point caused by a flexible strip load of width B. (See Figure 9.8.) Let the load per unitarea of the strip shown in Figure 9.tl be equal to q. lf we consider an elemental stripof width dr, the load per unit length of this strip is equal to q dr. This elemental stripcan be treated as a l ine load. Equation (9.15) gives the vertical stress increase do.atpoint,4 inside the soil mass caused by this elemental strip load. To calculate the ver-tical stress increase, we need to substitute q dr for 4 and (x - r) for r. So

do- -2(q dr)23

(e.17)n l ( x - r ) 2 + z z f '

The total increase in the vertical stress (Aa.) at point ,4 caused by the entire stripload of width B can be determined by integration of Eq. (9.17) with l imits of r from- Bl2 to + Bl2. or

Lo,: |

0",: l_)r'

z'- - \ 2 | - 2 1 2- t ) r 1 )

: 1{o"''lr:ol- tan- lr;aalB z l x 2 - z ' - 1 n ' t + 1 1

l * ' + z ' -z' - g't+11 I- - - /(B t l4 ) l t + B 'z ' )

( z q \ f\ ; / lG \ . ,

(e.18)

Table 9.3 shows the variation of L,o,lq with2zlB for 2xlB. This table can beused conveniently for the calculation of vertical stress at a point caused by a flexiblestrip load.

(e.

1 . 00.9

Ao

0t.,

7

r.000 0.0000.909 0.5000.'/75 0.5000.697 0.4990.6-51 0.4980.620 0.4970.-598 0.49.50.-5r3r 0.4920.566 0.4890.-552 0.,1850.,540 0.4800.529 0.4740.-5 I7 0.'{6u0.506 0.1620.495 0.4.550.484 0.4480.474 0.4400.463 0.4330.4-53 0.1250.443 0.4t70.433 0.4090.423 0.4010.413 0.3930.4(J4 0.38-s0.39-s 0.3780.3t36 0.3700.377 0.3630.369 0.3-5-50.360 0.34iJ0.352 0.3410.34-5 0.3340.337 0.3270.330 0.3210.323 0.31-50.3I6 0.3080.3 10 0.3020.304 0.2970.298 0.2910.292 0.213.50.286 0.2800.280 0.2750.275 0.2700.270 0.2650.265 0.2600.260 0.2560.2.55 0.2510.251 0.2470.216 0.2430.242 0.2390.238 0.2350.234 0.23r

0.8

1.0000.9800.9090.8330.7730.7270.69 r0.6620.6380 .6170.5980.-5u00.5640..54tt0.53't0. -5I90.5060.4920.1790.4670.45-50.'1430.4320.4210 .4 I00.4000.3900.3f1I0.3720.3630.3-550.3470.3390 .3310.3240 . 3 I 70 . 3 1 00.3040.2970.2910.2n.50.2t3t)0.2740.2690.2640.2590.2540.2500.2450.2410.237

0.7

1.0000.9930.9.590.9080.8-550.rJ080.7670.7320.7010.67-s0.6-s00.6280.6070.-su80.-56r)0.-5520.53-50 . .5 I I0..s040.4tt90.4750.4620.4490.4310.4250 .4 t40.4030.3930.3830.3730.3640.35-50.3470.3390 .3310.3230.3I 60.3090.3030.2960.2900.2840.2780.2730.2680.2630.2-580.2530.2480.2440.239

0.6

1.0000.9970.9790.9470.9060.8640.13250.7880.7.s50.1240.6960.67(')0.6460.6230.6020 .5810..5620.5440..5260 . 5 I 00.1940.4190.4650.4-510.43tt0.4260. '1140.4030.3920.3rJ20.3720.3630.3-540.34.50.3370.3290.3210 . 3 1 40.3070.30I0.2940.2880.2820.2760.27 L0.2660.2600.2-5,50.2510.2460.242

18)l

,- /B0.5

1.0000.9980.98u0.1)670.9370.9020.rJ660.8310.7970.7650.73-50.7060.6790.6-5'10.6300.6070.5n60.-56.50.5460.-s2t30 . 5 I 00.4940.4790.4640.4500.4360.4240.4t20.4(x)0.3u90.3790.3690.3fi)0.3.5 t0.3420.3340.3260.3 l r l0 .31 I0.3040.2980.21)10.28-50.2790.2740.2680.2630.2580.2530.2480.244

2xlB [Eq.

0 .4

1lnO0.9990.9920.9780.9550.9270.tt960.it630.8290.7970.7660.73.50.7070.671)0.6-530.6290.6050.5u30.-5630.5430.5210.5070.4900.4140.4600.44.50.4320 . 4 I 90.4070.3960.31350.37-s0.36-s0.35-50.3460.3380.3300.3220.31.50.3070.3010.2940.2880.2820.2760.2700.2650.2600.2-550.2500.245

lB and

0.3

I J000.9990.9950.9840.9660.9430 .91 50.88-50.8-530.8210.7890.7.580.7280.6990.6720.6460.6210.-s9f,i0.5160.55-s0..53-s0 .5 I 70.4990.4u30.4670.1520.4390.4250.4I 30.4010.3c)00.3790.3690.3590.3-s00.3410.3330.32.s0 .3170 .3 l 00.3030.2960.2900.2830.2'/80.2720.2660.2610.2560.2510.246

,lq with 2z

o.2

1J000.9990.9960.9870.9730.9.s30.9280.8990.8690.8370.8050.77 40.7430 .7140.6u50.65r10.6330.6080..5u-50.5640.-s430.5240.-5060.4890.4730.4580.4430.4300.4170.40.50.3930.3u20.3720.3620.3-520.3430.33.50.3270 .3 r90 . 3 r 20.3040.29it0.2910.28-50.2790.2730.2680.2620.2570.2520.247

tion of I

0 . 1

I f,001.0000.9970.9u90.9760.9-580.93-50.90[t0.u780.8470.8I -50.7t330.7520.7220.6930.6660.6390.6I -50. .5910.-5690.-54t30.5290.-s I00.4930.4760.4610.4460.4320 .4 I90.4070.3950.3u40.3730.3630.3540.34.50.3360.3280.3200 . 3 1 30.3050.2990.2920.2860.2800.2740.2680.2630.2-580.2530.248

0.9-580.93-50.90[t0.u780.8470.8I -50.7t330.7520.7220.6930.6660.6390.6I -50. .5910.-5690.-54t30.5290.-s I00.4930.4760.4610.4460.4320 .4 I90.4070.3950.3u40.3730.3630.3540.34.50.3360.3280.3200 . 3 1 30.3050.2990.2920.2860.2800.2740.2680.2630.2-580.2530.248

9.3 Varia

0.0

1J001.0000.9970.9900.9770.9590.9370 .9100.8810.8500 .81 80.7810.7.5.50.7250.6960.66tt0.6420.6t70.5930.-5710.5.500.,5300 .5 I I0.4910.417(\.4620.4470.4330.4200.40t30.3960.31350.3740.3640.3540.3450.3370.32r10.3200.3 l30.3060.2990.2920.2860.2800.2740.2680.2630.2580.2530.248

lll'lies| 3.80I 3.e0| 4.00

I 4.10| 4.20

| 4.30

| 4.40

I 4.50I 4.604.704.804.905.00

(continued)

Table 9.3 (continued\

2x /B

1 . 91 . 81 . 71 . 61 . 51 . 41 . 31 . 22zl B 1 . 1

0.000 . 1 00.200.300.400.500.600.71)0.800.90t . (x)l . l 01 .201 .301 .401 .501 .60l . '70l . l r01 .902.(X)2 . 1 ( l2.202.302.402.502.602.702.rJO2.1)o3.(X)3 . l 03.203.303.403.-503.603.703.803.904.(X)4 . 1 04.204.304.404.504.6i)4.704.804.905.00

0.0000.0910.2250.3010.3460.3730.3910.4030.41 I0.41 60.4I 90.4200. ,1I9o.4t70.4t40.41 I0.4070.4020.3960.3c) I0.38-50.371)0.3730.3660.3600.3540.3170.34I0.3350.3290.3230 . 3 I 70 .31 I0.3050.3(x)0.21)40.2890.2840.2'790.2'740.2690.2640.2600.2-5-50 .2510.2470.2430.2390.2350.2310.227

0.0000.0200.09 r0.16-50.2240.26'70.2980.3210.3380.35 r0.3600.3660 .3710.3730.374o.3740.3730.3700.3680.3640.3600.3.560.3,520.3410.3420.3370.3320.32'70 .3210 . 3 I 60 .31 I0.3060.30I0.2960.2910.2IJ60.21310.2760.2120.2670.2630.2.5u0.2540.2500.2460.2420.2380.2350.2310.2270.224

0.0000.0070.0400.0900 . 1 4 10.18.50.2220.2-500.2730.2910.30-50 .3 r60.3250 .3310.33.50.33rJ0.3390.3390.3390.33fJ0.3360.3330.3300.3210.3230.3200 . 3 I 60 .3 r20.3070.303o.2990.2940.2900.2860.2u 10.2710.2730.26r10.2640.2600.2560.2520.2480.2440.2410.2370.2310.2300.2270.2230220

0.0000.0030.0200.0-520.0900 .1280 .1 630 .1 930 .2 r80.2390.2560.27 |0.2820.2910.29tt0.3030.3070.3090 .3 I I0 . 3 I 20 .3 I I0 .3 I I0.3090.3070.3050.3020.21)90.2960.2930.2900.2u60.2u30.2790.2150.2110.2680.2640.2600.2560.2.530.249(\.2460.2420.2390.2350.2320.2290.2250.2220.2r90.216

0.0000.0020 .0110.0310.0590.0u90 . 1 2 00 .1480. t '730 .1 950.2t40.2300.2430.2540.2630.2110.2760.21310.2810.2rJ60.281J0.2880.2i380.2t{30.2870.21J.50.2tt30.2u 10.2790.2760.2740.27 |0.2680.2650.2610.2580.2-550.2520.2490.2450.2420.2390.2360.2330.2290.2260.2230.2200.21'70.2150.2t2

0.0000.0010.0070.0200.0400.0630.0880 . 1 l 30. t370 .1 5u0 . t770. t940.2090.221o.2320.2400.24u0.2540.25t30.2620.2650.2610.26u0.26u0.2680.268o.267o.2660.2650.2630.2610.2-590.256o.2540.2510.2190.2460.2430.2400.23rJ0.2350.2320.2290.2260.2240.2210 .21 80.2150.2130.2 r i)0.207

0.0000.00r0.0040 .0 r30.0270.0460.0660.0870 .1080 .1 280. t4 '70 .1640 .1 7u0 . 1 9 10.2030 .2130.2210.22u0.2310.2390.243O.24(t0.24r10.2-500.2-510.2-510 .2510.2510.2500.2490.2480.2470.2450.2430.2410.2390.2310.2350.2320.2300.2270.2250.2220.2200.2t70.2150.2t20.2100.2080.2050.203

0.0000.0000.0030.(x)90.0200.03,10.0500.0680.0860 .1 040 .1220 .1380 . r520 .1 660 . t710.Itrtt0.lL)10.20-so .2 t20 . 2 t 70.2220.2260.2290.2320.2340.235('t.2360.2360.2360.2360.2360.23.50.2340.2320.2310.229(\.2280.2260.2240.2220.2200 .21 80.2160.2130.2110.2090.2070.2050.2020.2000.19rJ

0.000 0.0000.000 0.0000.002 0.0020.007 0.0050 .014 0 .0110.02-5 0.0190.038 0.0300.053 0.0420.069 0.0560.0135 0.0700.101 0.0840.1 l6 0.0980 .130 0 .1 I 10 .143 0 .1230. I -55 0. 1350. r 65 0.1460. I 7.5 0. 1550 .1n3 0 .1640 . 1 9 l 0 .1720 .197 0 .1790.203 0.ltt50.20n 0.1900 .2 t2 0 . r 950.21 -5 0. I 990.2t7 0.2020.220 0.2050.22t 0.20'70.222 0.2080.223 0.2100.223 0.2t10.223 0.2il0 .223 0.2120.223 0.2120.222 0.2110.221 0.2110.220 0.2100.21[t 0.2090.217 0.2080.2t6 0.2010.214 0.2060.2t2 0.2050.211 0.2030.209 0.2020.207 0.2000.20-5 0.1990.203 0.1970.20t 0.1950.199 0.1940.197 0.7920.195 0.1900.193 0.188

9.5 VerticalSfress Due to Embankment Loadina

Example 9.4With reference to Figure 9.8, we are given q : 20AkN/m2, B : 6m, and e : 3 m.Determine the vertical stress increase at J : +9, +6, +3, and 0 m. plot a graph ofAo" asainst -r.

E I 2 0zJ-

d ' n 0

- t ( , - t r _o _ ,+ I 0 2. \ ( n l )

Solut ionThe lollowing table can be made:

x(m) 2xl B 2zl B

"From Table 9.3i'4 : 200 kN/m2

The plot of Arr, against x is given in Figure 9.9.

Figure 9.9Plot of Ao- againstdistance x

a,uJq" Ao," (kN/m2)+ ( , + ? |

+ 6 + ) l

+ f + l t

1 ) n l

0.0170.0840.4i100 .818

J . +

16 .896.0

r63.6

9.5 Vertical Sfress Due to Embankment LoadingFigure 9. l0 shows tlrc cross scctior.r of an cmbankment of height H. For this two-dimcnsional Ioadins condi t ion the ver t ica l s t ress increase maV bc expressecl as

/ J r - | + 8 , + l

Figure 9.10Embankment loading

238 Chapter 9 Sfresses in a Soil Mass

where q,,

vH

Figure 9.11Ostcrberg'.s chartfor determinationof vert ical stressdue to embank-ment loading

Ao. = +VLt')t*, * "; - fra,tlyHunit weight of the embankment soilhcight of the embankment

a , l r a d i a n s ) - - t a n ' ( B ' ! - B ' )

- , o n\ i , /

, / B , \0 . : t a n t I-

\ 2 , /

(?)

(e.1e)

(e.20)

(e.21)

3.02 .0t . 6t . 4t . 2

t . 0

0 .9

0 .8

0.1

0.6

9.5 Vertical Sfress Due to Embankment Loading 239

For a detailed derivation of the equation, see Das (199'7). A simplif iecl form ofEq. (9.19) is

Lo r. : q,,[2 (e.22)where 1, : a function of B1lz and B2lz.

Theva r i a t i ono f l rw i t h B l l z .andB2 lz , i sshown inF igu reg . l l ( os te rbe rg , l 957 ) .

Example 9.5

An embankment is shown in Figure 9.12a. Determine the stress increase underthe embankment at points A, and A2.

Solution

yH : (17.s)(7) : 122.5 kN/m2

Stress Increase at A,The left side of Figure 9.12 indicates that Br :2.5 m and 82: l4 m. So

B t : z s - : 0 s . 4 : - l j : r oz 5 z 5 - " '

According to Figure 9.11, in this case, Iz = 0.445. Because the two sides in Fig-ure 9.12b are symmetrical, the value of 1, for the right side will also be 0.445. So

Lor.: Ao411 * Lor.121 : erllzg-"try * Iz(nieh,]

- 1"22.510.445 + 0.4451 - 109.03 kN/m2

l a In -J+-5rn > l+- 1 .1 r '+ l-+- -- "" ...t ' - . .t . .

.".-oJo H =r7 m

^,,o.t' { y= tl.S kNlm: -''',.,.,.

. : " . . 1 1 ' : ; ' ; - . 1 , . . f j , - - r : . . t . . ' : ' ' t , . . . - f . ' ' . , . . . , ' _ . r : , . . ' . . , t . . , . " . . . , . . ' 1 . , : ; . . _ .i - t - t - l

- \ m - 5 ' t n l l . 5 l i l 5 n r 1 6 . 5 r r r

2.-5 rl

l - * l .+ l '1 m+l

IIi

_ " : l | ) VaA l

l<- lam+F-+l

Figure 9.12

Chapter 9 Stresses in a Soil Mass

| . - t

" t , l .1.) = (2.-5 m)x ( 17 . -5k N / m r ) =

I '13.75 kN/mrI

II

Y Ao. ( r )aA .

Thus, /2 : 0.495. So

For the right side

( 7 m ) x( 17.-5

kN/mr )= l ) r 5

kN/rr l

Ao,ot: 0.495(122.5) : 60.64 kN/m'

I

] . - . r ,u - - - - ]d \ , r t r y

a

( c ) A '

Figure 9.12 (continued)

Stress Increase at ,42Refer to Figure 9.12c. For the left side, 82 : 5 m and 81 : 0. So

B z 5 B t 0 ^

z 5 ? , 5

According to Figure 9. I 1, for these values of B2l z and B 1l z, /z : 0,25. S<r

Ao.1r1 : 43.75(0.25): 10.94 kN/m2

For the middle section.

1 4- ' ' - a o- - - L , O

8 2 1 4 ^ ^ B '

z . 5 z

8 2 g - ^ B t 0- - - - l X ' : - : 07 . 3 2 )

and 12: 0'335' So

Ao.(:) = (78.75X0.335) :26.38 kN/m'

Total stress increase at point ,4, is

Ao , : Ao . r r r * Lo r ( r , - Aa , r s ) : 10 .94 + 60 .64 - 26 .38 :45 .2kN /mz

l<- 1,1 nr -=-+-l.r t+ nr ---l

9,6

9.6 vertical stress Below the center of a LJniformty Loaded circular Area 241

Figure 9.13Vcrtical stress bclow thc centcr of a unil irrmly loadcdf lcx ib le c i rcu lar area

VerticalSfress Below the Center ofa Uniformly Loaded Circular Area

Using Boussinesq's solut ion for ver t ica l s t ress Ar . r - caused by a point load IEq. (9.12) ] ,one can also devclop an exprcssion fclr thc vertical stress bektw the center of a uni-formly loaded f lcx ib lc c i rcu lar area.

From Figurc 9. 13, lct thc intensity of pressurc on thc circular area of radius Rbe equal to q. Thc total loerd on the elcmental area (shaded in the figure) is equal toqr dr da. The vertical strcss. rlrr- at point A caused by the load on the elemental area(which mery be assumed to be a concentrated load) can be obtaincd from Eq. (9.12):

The increase in the stress at point A caused by the entire loaded area can befound by integrating Eq. (9.23):

, 3(q, dr dtt) 2.3r i r r - . -

_ , ,' Ztr (r, + z2)s,2

I t , , 2 n S r l t 7 , , _ . \ , "

A o . - l d o , - | | : - ' , 4 r t r d aJ J , , 0 J , , t l 7 r ( r ' * z ' )

(e.23)

So

L o , : n { t -l (Rlz)z + 1)3tz

(e.24)

Chapter 9 Sfresses in a Soil Mass

Table 9.4 Variation of Lc-"lq with e/R [eq. (9.2a)]

zlR Lc,

10.99990.99980.99900.99250.94880.91060.75620.64650.42400.2u450. I 9960. I 4360.08690.0571

Figure 9.14Stress under the center of a uni-fornrly loadcd flexihle circular area

Thevar iat ionof L,o, lq wi thz/Rasobta incdfromEq. (9.2a) isg iveninTable9.4.A plot of this is also shown in Figure 9.14. The value of Aa- decreases rapidly withdepth, and at z :5R, it is about 60/o of 4, which is the intensity of pressure at theground surface.

Vertical Sfress at any Point Belowa Uniformly Loaded Circular Area

A detailed tabulation for calculation of vertical stress below a uniformly loaded flex-ible circular area was given by Ahlvin and Ulery (1962). Referring to Figure 9.15, wefind that Lo, at any point ,4 located at a depth z at any distance r from the center ofthe loaded area can he s iven as

A o , : q ( A ' + B ' ) ( g . 2 5 )

where A' and B' are functions of zlR and rl R. (See Tables 9.5 and 9.6 on pages 244and245.)

Vertical Sfress Caused by aRectangularly Loaded Area

Boussinesq's solution can also be used to calculate the vertical stress increase belowa flexible rectangular loaded area, as shown in Figure 9.16. The loaded area is lo-cated at the ground surface and has length L and width B. The uniformly distributed

00.020.0.50 . 1 00.20.40.50.8t . 01.-52.02.53.04.05.0

9.7

9.8

I

" " ' -: " " " t q * " 1 b r .* " " ' . r d ; ' . . 1 o r " f . . . . '' , - , i . '

l u . "Y . " ' . ,1* " . , " ;1 . , o ' " .: :.1:: :l ,f.Tf,',;-* . - t l t - l - ' * t :' .

" , ; " . I "" . tr ." ", ' -_

r| - -t - -It -l i .

9.8 Vertical Sfress Caused by a Rectangularly Loaded Area

Figure 9. 15 vert ical stress al any point below a unitbrmly loaded circular area

lo,,I

A-1w

tooI

IIIY .

Figure 9. 16 Vertical stress below the corner of a uniformly loadecl flexiblerectangular area

load per unit area is equal to 4. To determine the increase in the vertical stress (Ao.)at point,4, which is located at depth z below the corner of the rectangular area, weneed to consider a small elemental area dx dy of the rectangle. (This is shown in Fig-ure 9.16.) The load on this elemental area can be given bv

d q : q d x d y (e.26)

Table 9.5 Variation of A' with zlR and rlR"

1 . 51 . 20 .6zlR

0 . 10.20.30.40.-s0.60.10 .E0.9It . 21 . 52l . )

3456l8

l 0

1.0 l . i )0.9(n50 0.897180.1t0388 0.798240.71265 0.70-5180.6286 1 0.6201 50.55279 0.-s4.+030.41t.550 0.'176910.,126-5,1 0.41 8740.3753r 0.36S320.33 101 0.324920.292t39 0.2117630.23 17rJ 0.227950. | 679.s 0. 165520. I0557 0 . I0 :1530.07I52 0.0709ri0.0.s 1 32 0.0.5 1 010.029,s6 0.029160.0I 9.12 0.0I 93,S0.0 I 36 I0.0I (x).s().ool12

0.(x)6 I2

1 . 0 1 . 00.88679 0.U61260.77u8.+ 0.734t330.6t33 16 0.626900.59241 0.537670.5 1622 0.464480.45078 0.401270.39491 0.35,121.t0.34729 0.3 12430.30669 0.211070.27(X).s O.24691t).21662 0.19ti900.15877 0. I4ft040. 1 01 40 0.096'170.069.17 0.0669ii0.0.5022 0.048ti60.02907 0.02n02

1.0 0..50.78797 0.4301 50.630 l4 0.3tr2690.520ti I 0.343750.44329 0.3104u0.38390 0.21J l-560.33676 0.2-55t{r0.291J33 0.211210.265n1 0.212970.23832 0. Ig,llJlJ0.2 146u 0. 17u6lJ0 .1 '7626 0 . t5 l0 l0 .13436 0 . I 18920.0901 | 0.0ri2690.06373 0.0597'+0.04707 0.044,s70.021132 0.02719

0.0I lJ350.0I 3070.()()9760.()()7.s.50.(x)f ix)

0 0 00.096.+-5 0.02'787 0.008-560.15433 0.052-s 1 0.016800.119(t4 0.07199 0.024400. I tt709 0.0t3-s93 0.03 I l80. I rJ556 0.09499 0.037010 . 1 7 9 5 2 0 . 1 0 0 1 0o.17 t24 0. l022rJ 0.045580.16206 0 .102360. l 5253 0.100940. I4329 0.09849 0.05 I850. 1 2.570 0.091 92 0.0.s2600. I 0296 0.0rJ0.+rJ r).05 1 160.0711| 0.0627-s 0.014960.055-s5 0.0413ft0 0.037tt70.04241 0.03E39 0.031500.026.s | 0.02490 0.02193

0.0I -5730.0I I6tt0.()()IJ94( ).(x)7030.(x)566

o.o011l 0.(x)465

' r ' A l ' t c r Ah l v i n anc l I J l c r v ( 19 ( r2 )

Table 9.6 Vrlialiorr of 1J' with z.l Il '.rnd rl R't'

r/R

1 . 51 . 20.80 .60 .4o.2zlB

r.)0 . I0.20.30.40 .50.60.70.tr0 .9

0 00.0(lri52 0. I0 t.100. lll,S57 0. I 93(Xr0.26362 0.261E70.320I 6 0.322.s90.35111 0.357520.371t31 0.37.s3 I0.3E'187 0.37t)620.31J09I 0.37,101J0.36962 0.362750.3.s355 0.3.15.s30.3I185 0.307300.2-s6()2 0.2502-s0. l 7lJlJg 0. l tl l,140 . I2807 0 . I26330.09487 0.093940.05707 0.056660.03172 0.03760o.026660.01 9800.015260.01212

t ,

0.1 I lS rJ0.201720.2n0I rJ0..127'+fi0.351230.3630fi0.360720.35 1330.337340.320750.28,+,S I0.2333n0. 1664,10.t21260.090990.05562

00. I3,1240.235240.29,1It30.322730.33 I(Xr0.32E220.3 19290.3(b990.2929t)0.2781 90.2zlti360.2069.10 . l -519n0. I I3270.0n6350.05383

00. I f i7960.2-59830.21251o.269250.262360.25.1I 10.2'+6380.231190.22n910.21 9780.20 I I30. t 736ii0. l 33750. | 02980.080330.05 I45

0 0 0 00.053,s8 0.07u99 0.02672 0.(X)tt450.0ti .sl3 (t .O1759 .0.044,11J 0.()15930. I 0757 0.043 | 6 0.0.1999 0.02 I660. l2'104 0.(X)76(r 0.04535 0.025220. | 359 l 0.02 I 65 0.03455 0.0265 I0. I44,10 0.04.1-57 0.02 I0 I0. l49ti6 0.0620(.) 0.(x)702 -0.02329

0. 1 5292 0.07530 0.(X)6 140. I5404 0.0t i507 0.0I7950. I 53.s5 0.(D2 I 0 0.0281 4 0.0I 0050. 14915 0. l(XX)2 0.0,+3713 0.(Xn230.1 3732 0.1 0 1 93 0.05745 0.01 3850. I 1331 0.09254 0.r)6371 0.028360.09130 0.07869 0.06022 0.034290.07325 0.0655 1 0.05354 0.0351 10.01713 0.04532 0.03995 0.030660.03384 0.024140.02.168 0.019680.0lrJ6l3 0.015770.01459 0 .012790 . 0 1 1 7 0 0 . 0 1 0 5 1

0.00921 0.00879

2-5

2.534567at

l 0

*Source: From "Tabulatcd Values lor Determining the Ciomplete Pattern ofStresses. Stra ins. and Def lect ions Beneatha Uniform Circular Load on a Honroseneous Half Space." b1' R. G Ahlvin and H. H. Ulery. ln Highway ResearchBulletin.342, Transportation Rt:scarch Board, Ni,rtional Research Council. Washinston. DC. 1962.

244

_ - ! q

(continued)

1 41 21 000.000840.001 670.002-50

(,0.000.120.00083

TJ U

0.000200.00048 0.00030

0.00118 0.(xx)71 0.0(x)53 0.00025 0.(n014 ().000090.00,+07 0.00209

0.(x)761 0.003930.00u7t 0.004590.01013 0 . (x )5480.0 t 160 0.00659o.0t22t 0.007320.01220 0.(x)7700.01109 0.(x)761J0.(x)949 0.(x)70rj0.(x)795 0.(x)6280.(x)66 t 0.(x)541J0.(x).s54 0.001120.(X)46(r 0.(X)4(X){).(x)397 0.(x)352

0.00226 0.(x) 143 0.000970. (x )269 0 .00171 0 . (x )11-s0.(x)32-s 0.(x)210 0.(x) 1110.(x)399 0.(X)264 0.(x) l u00.(x)463 0.(x)301J 0.(x)2 I40.(x)50.s 0.(x)346 0.00242().(x)-s36 0.(x)3u4 0.(x)2rJ20.(x).s27 0.(x)394 0.(X)2980.(x)492 0.(x)31t4 ().(x)2990.(x)445 0.(x)36() 0.(x)29 |0.(x)39u 0.(x)332 0.(\ t)2160.(x)3.s3 0.(){)301 0.(x)2560.(X)326 0.(X)273 0.(X)2,11

0.0(x).50 0.0(x)29

0.(xx)73 0.(xx)430.(xx)94 0.(xx)-560.(x)t l 5 0.(xx)6ti0.(x) 1 32 0.(x)()790.(x) lf i) 0.(xx)990.(x)179 0.(x) l 130.(x) lult 0.(x) 124() . (x)193 0.(x)1300.(X) 1 lJ9 0.(X) l 3;10.(x) lt i4 0.()() 133

0.0(x) I rJ

0.o('i0270.(xx)360.(xx)430.(xx)510.(xx)6.50.(xx)750.(xx)u40.(xx)910.(xx)940.(xxD6

( t t t t t t i t r t r t ' t l )

1 41 21 00 0 00.()(x)ti4 -0.(xx)42

--0.(x)166 O.(xxllJ3 0.(xx)2,10.(x)245

0.(x)01 5 O.(xx)l 0

0.(xx)73 ,-0.(xx)49

-0 .01115 0 . (x )608-0.0099-s -0.(x)632-0.00669 0.006(x)0.0002ti *0.(x)410

0.0066 1 0.001 300.01 112 0 .001570.015 15 0.0059-50.01522 il.00u l t)0.01380 0.008670.01204 0.008420.01034 0.007190.00B88 0.0070.50.00764 0.00631

-0.(n3.+,t -0.00210

0.(x)378 0.002360.0040I -0.(x)265

-0.00371 0.0027rJ0.0027 | -0.002-s0

-0 . (x )134 -0 .00 t92

0.00155 0.000290.0037 I 0.001 320.00496 0.002-540.00547 0.003320.00-551 0.003720.00533 0.003860.00-501 0.00382

0.(x) I 3-5 0.000920.001-s6 0.(x)107

-0 . (x ) t8 t -0 . (x )126

0.00202 0.()0l4lJ-0 .00201 -0 . (D156-0 .00r79 0 . (n l5 l

0.0009.1 0.001090.00013 -0.00043

0.001 l0 0.000280.t)01 85 0.000930.00236 0.001410.00265 0.001780.00281 0.00199

0. (x )3 t i8 0 . (x )199 0 . ( ) ( ) t t6 0.(xx)2-s 0.(xn l4 0.00(x)9

0.00041J 0.00021r - 0.0001 u

-0.00068 0.000,+0 0.000260.(xx)84 -0.(xD-50 -0.00033

-0.00094 -0.00059 0.000390.0(x)99 0.0006-5 - 0.000160.00094 -0.(x)068 -0.00i)50

-0 . (x l { }7u u . (x }061 0 0(x t4o0.00037 0.000.+7 0.0004-s0.00002 ,0.00029 -0.000370.00035 -0.0000u -0.000250.00066 0.00012 0.00012

245

246 Chapter 9 Sfresses in a Soil Mass

The increase in the stress (do,) at point ,4 caused by the load dq can be determinedby using Eq. (9.12). However, we need to replace Pwith dq: q dx dy and I withx2 + y2. Thus.

d o , :r l

Jq ox oy z ' (e.27)2 r r ( x 2 + y ' + z ' ) t ' '

The increase in the stress, at point A caused by the entire loaded area can now bedetermined by integrating the preceding equation. We obtain

I fB 1L Zqz3(dx dy)A o ' : J o o ' : J , - . L - , f f i " : t r '

/ m 2 + n 2 + 2 \ , / z ^ r x 6 r f + i + t \I l + l a n ' l . . ^ ^ |\ m + n ' + l / \ m ' + n ' - m ' n ' I l /

(e.28)

where

- Bz

: LZ

(e.2e)(e.30)

(e.31)

The variation of 1., with m and n is shown in Table 9.7.

Table 9.7 Variation of 1., with m ancl n [Eq. (U.29)l

1 . 00.70.60.40.30.1 0.5

0 .10.20.30.40.50.60.70.80.91 . 01 . 21 . 41 .61 .82.02.53.04.05.06.0

0.0047 0.00920.0092 0.01790.0132 0.02590.0168 0.03280.0198 0.03870.0222 0.043.50.0242 0.04740.0258 0.05040.02'70 0.05280.0279 0.05470.0293 0.05730.030r 0.05890.0306 0.05990.0309 0.06060.0311 0.06100.0314 0.06160.031-5 0.06180.0316 0.06190.0316 0.06200.0316 0.0620

0.01320.02590.03740.04740.0-5-590.06290.06860.07310.07660.0'7940.08320.08560.08710.08800.08870.08950.08980.09010.09010.0902

0.016t30.03280.04740.06020 .071 I0.08010.08730.09310.09'770 .10130.1 0630.10940 . 1 1 1 40.11260 . 1 1 3 40.1 1450 . r 1500 .11530.11540.1154

0.0198 0.02220.0387 0.04350.05-s9 0.06290.0711 0.080r0.0840 0.09470.0947 0.10690.1034 0.1 1690.1104 0.12470 . 1 1 5 8 0 . 1 3 1 10.1202 0.13610.1263 0.14310.1300 0.t4750.t324 0.15030.1340 0.15210.1350 0.15330.1363 0.1s480.1368 0.15550.1372 0.15600.1314 0.15610.7314 0.1562

0.0242 0.025rJ0.0474 0.05040.0686 0.07310.0873 0.09310.1034 0.r 1040 .1168 0 .12470.1277 0.13650 .1365 0 .146 r0.1436 0.15370.1491 0.1-5980.1.570 0.16840.1620 0.17390.1652 0.17740.7672 0.17970.1686 0.18120.1704 0.18320 .1711 0 .18410.r7t7 0.18470.1719 0.18490.17 t9 0.1850

0.0270 0.02'790.052u 0.05470.0766 0.0"/940.097'7 0.10130.1 t 58 0.12020 . t31 I 0 .13610.1436 0.14910.1537 0.15980 . r619 0 .16840.1684 0.17520.17'77 0.18510.1836 0.19140.18'74 0.19550.1899 0.19810.1915 0.19990.1938 0.20240.1947 0.20340.1954 0.20420.1956 0.20440.t957 0.2045

9.8 VerticalStress Caused by a Rectangularly Loaded Area 247

Figure9.77 Increaseof s t rcssa tanypo in tbe lowarec tangu lar lv loadedf lex ib learea

The increase in the stress at any point below a rectangularly loaded area canbe found by using Eq. (9.28). ' I 'his

can be explained by reference to Figure 9.ri.Letus dete rmine the strcss at a point below point ,4' at depth z. The loaded area can bedivided into four rectangles as shown. The point A' is the corner common to all fourrcctangles. The increasc in the st rcss at depth z below point .4 'due to each rectan-gular area can now be calculated by using Eq. (9.2t3). The total stress increase causedby the entirc loaded arca can be given by

Arr- : qll.r,l + |.2; * 11ry + 1.,t.t] (e.32)

where 1.( ,), 1.,(rr,1.,,3,, and 1.,,., : values of / j for rectangles 1,2,3, and 4, respectively.

(continuctl)

5.03.02 .52 .O1 . 81 . 61 . 4

3

2 1

A '

L

0.08320.10630.12630,143 I0.15700.1 6840.17710.1 85 I0.19.s80.20280.20730.27030.21240,21510.21630.21720.2r'750.2176

0.030I0.05t390.0u-560. 109,10. r 3000. r 47.50.16200. I 7390 .18360 . I 9140.20280.21020 .21510.21830.22060.22360.22500.22600.22630.2261

0.031 I0.06 t00.0ttu70 . 1 1 3 40. I 3500. I 5330. I 6860. l f : i l20 . 1 9 1 50. I 9990.21240.22060.22610.22990.232s0.23610.23780.23910.23950.2397

0 .03 I60.061 90.09010. t l -530 .13720. 1.5600 . 1 7 t 70.1 8470. I 9540.20420.21720.22600.23200.23620.23910.24340.24550.24720.24790.2482

0.03()6 0.03090.0599 0.06060.0u7 t 0.08800 . l l l ' 1 O . 1 1 2 60. t321 0.13400.1503 0.15210.16-s2 0. t6720 . t174 0 . t 1910 .1874 0 .18990 . I955 0 . I 9 t J I0.2073 0.2i030.21.51 0.21840.2203 0.22370.223'7 0.22740.2261 0.22990.2294 0.23330.2309 0.23500.2320 0.23620.2324 0.23660.2325 0.2367

0.03 l4 0.03 I .50.06r6 0.06tr l0.0t39-5 0.0u9u0 . t l 4 -5 0 .11500.1363 0. l36t t0.L54tt 0.L5550.1704 0.11110 .1832 0 .18410. 1938 0.19470.2021 0.20340.2151 0.21630.2236 0.22500.2294 0.23090.2333 0.23500.2361 0.23780.2401 0.24200.2420 0.24390.2434 0.24550.2439 0.24600.2441 0.2463

0.0316 0.03160.0620 0.06200.0901 0.09020. I 1-54 0. I 1540.137 4 0.137 40. 1.561 0.15620.17l9 0.17190.1849 0.18500. 1956 0.1 9-570.2044 0.20450.2175 0.21760.2263 0.22640.2323 0.23250.2366 0.23670.2395 0.239'70.2439 0.24410.2461 0.24630.2479 0.24810.2486 0.24890.2489 0.2492

248 Chapter 9 Sfresses in a Soil Mass

I

I Ao'In1

?:lLl A oI

III

Figure 9. 18 Ve rt ical slr-ess bclow thc ccntcr ol a

uni l 'orrnly Ioacle d l lcxiblc rcctanguli tr arclt

In ntost cases the vcr t ica l s t ress incrcase below the center o l 'a rcctangular area

(Figurc 9. l t3) is important . This s t rcss incrcase can bc g iven by the re lat ionship

L , t r - : q l a (e.33)

wherc

n l t n I (e.34)

(e.3s)

(e.36)

(e.37)

n l y l 1 - * 1 1 4 6n +if1 + ,fn1 t2

t 1

I

, n t : E

Z.t t I

D

Bb :

2

The variation of -Ia with rz, and n, is given in Table 9.8.

9.8 Vertical Stress Caused by a Rectangularly Loaded Area 249

Table 9.8 Variation of 1o with rn, and n1 [Eq. (9.3a)l

0.20 0.9940.40 0.9600.60 0.ft920.u0 0.u001.00 0.701t .20 0.6061.40 0.5221.60 0.449l .u0 0.3uu2.00 0.3363.(X) 0.1794.(X) 0. lOt t-5.(X) 0.0726.(X) 0.0517.00 0.038il.(x) 0.0299.00 0.023

10.(x) 0.01 9

0.997 (\.997 ().9970.977 0.977 0.9770.936 0.936 0.9370.87E 0.tt80 0.ti8I0 .u14 0 .817 0 .8 l t t0.748 0.7.53 0.7-540.68-s 0.692 0.6940.627 0.636 0.6390.-s73 0.-5tt5 0.-s900.-s2-5 0.-s40 0.-5450.34ti 0.373 0.3u40.211 0.269 0.2n50.1 ' / 4 0.202 0.2190. I 30 0. l -s-s o.n20 . l ( x ) 0 .122 0 .1390.079 0.0911 0.1 l30.064 0.0u 1 0.0940.053 0.061 0.079

0.1)97 0.9910.91'/ 0.9110.937 0.9370. t381 0.8810.818 0. t i l 80.7-s-s 0.75-50.69-5 0.69-50.6,10 0.6410..591 0.-5920.517 0.-5480.3u9 0.3920.293 0.29u0.229 0.2360. l t t4 0. t920 . I 5 0 0 . I 5 u0 .12 -s 0 .1330 . 1 0 - s 0 . 1 1 30.089 0.097

0.997 0.997 0.99'70.977 0.1)7'7 0.97'70.937 0.937 0.9370.881 O.utt 1 0.uf]lO.ft I ft 0.ti i l8 0.8 t 80.7.s,5 0.7-5-s 0.75-50.696 0.696 0.6960.641 0.641 0.6420.-592 0.-593 0..5930.-s49 0.549 0.5490.393 0.394 0.39.s0.301 0.302 0.303(\.240 0.242 0.2440.197 0.2(x) ( 't.2020 . 1 6 4 0 . 1 6 8 0 . 1 7 10.139 0.114 ( \ . t470 .1 l 9 ( \ . 124 0 . t 280 . 1 0 3 0 . 1 0 u 0 . 1 l 2

0.9970.9160.9320.8700.u(x)0.7270.6-su0.-5930.53'+0.4u 10.2930 . I900 . 1 3 10.09.50.0120.0-560.04-50.037

I3 n t

I

Example 9.6

The flexible area shown in Figure 9.19 is uniformly loaded. Given that q =150 kN/m2. determine the vertical stress increase at ooint A.

| -5 rr = radius

A

r/ - 150 kN/rnr

Figure 9. t9 Uniformly loaded flexible area

Chapter 9 Sfresses in a Soil Mass

l4.---8 m >lao.r2r

Ao:r l )

A

l - sm+ l

Arca 3 Figure 9.20Division of uniformly loaded flexible areainto three par ts

-f

1 . 5 n rI

-f

.5 rnI

SolutionThe flexible area shown in Figure 9.19 is divided into three parts in Figure 9.20.AIA,

Ao, * Ao.1ry + Ao12; t Acr.1:y

From Eq. (9.24),

Ao,i l: (;) ,{'- *ar;r"1We knowthat R = 1".5 m, z * 3 m, andq : 150kN/m2, so

Ao. ( ' : T { ' ia . " iu " } - , '3 kN/m2

We can see that L,aap1* Ao.(r).From Eqs. (9.30) and (9.31),

1 . 5m - : :* : O'5

8, :

, : 2 . 6 1

From Table 9.7, for m:0.5 andn:2.67,the magnitqde of 13 = 0'1365. Thus,from Eq. (9.28).

-

.L'o,Q): Ao.(r) = ql, - (150X0'1365) :20'48kN/m2

so

Ao. : 213 +2A.48 + 20.48 : 62.26 kN/m2 I

4 :z

9.9 lnfluence Chart for Verticat pressure

Influence Chart for Vertical Pressure

Equation (9.24) can be rearranged and written in the form

251

g,g

(e.38)

Note that Rlz and A,o -lq in this equation are nondimensional quantit ies. The valuesof R/z that correspond to various pressure ratios are given in Table 9.9.

Table 9.9 Yalues of Rlz. for Various pressure Ratios [Eq. (9.3g)]

Ao2lq Rlz L,o2lq Rlz

U0.0-50 . 1 00. l -50.200.250.300.3.50.400.4-50.50

U0. 186-50.26980.33830.400-50.459[J0.-5I t3 I0.576{30.63700.69910.7664

0.5-50.600.6-s0.700.7-50.u00.8.50.900.951 .00

0.u3840.91761.00671 .10971.2328L3871l.-59431.90tt42.5232oo

Using the values of Rlz obtained from Eq. (9.38) fbr various pressure ratios,Newmark (1942) presented an influence chart that can be used to determine the ver-tical pressure at any point below a uniformly loaded flexible area of any shape.

Figure 9.21 shows an influence chart that has been constructed by drawing con-centric circles. The radii of the circles are equal to the R/z values corresponding toL o " l q : 0 , 0 . 1 , 0 . 2 , . . . , 7 . ( N o t e : F o r L , o , l q : 0 , R l z : 0 , a n d f o r A , o . l q : 1 , R / z j o o ,

so nine circles are shown.) The unit length for plotting the circles is AB. The circlesare divided by several equally spaced radial l ines. The influence value of the chart isgiven by 1/N, where l/ is equal to the number of elements in the chart. In Figure 9.21,there are 200 elements; hence, the influence value is 0.005.

The procedure for obtaining vertical pressure at any point below a loaded areais as follows:

1. Determine the depth z below the uniformly loaded area at which the stress in_crease is required.

2. Plot the plan of the loaded area with a scale of z equal to the unit length oft he cha r t (AB) .

3. Place the plan (plotted in step 2) on the influence chart in such a way that thepoint below which the stress is to be determined is located at the center of thechart.

4. count the number of elements (M) of the chart enclosed by the plan of theloaded area.

252 Chaoter 9 Stresses in a Soil Mass

Influencevalue = 0.005 Figure 9.2? Inl luence chart l i rr vert ical

pressurc hased on Boussincsq'.s thcttry(a f tc r Newmark . 1942)

The increasc in the pressurc at the point undcr consideration is given by

Lo , : ( lV )qM

where 1V : in f luence valuer/ : pressure on thc loaded arezr

(e.3e)

Example 9.7

The cross section and plan of a column footing are shown in Figure 9.22. Find theincrease in vertical stress produced by the column footing at point A.

SolutionPoint,4 is located at b depth 3 m below the bottom of thq footing. The plan of thesquare footing has been replotted to a scale of AB -{3 m and placed on theinfluence chart (Figure 9.23) in such a way that point.4 bn the plan falls directlyover the center of the chart. The number of elements inside the outline of the planis about 48.5. Hence,

/ 6 6 0 \Lo, = (rv)qM = 0.00s(;;

)+s.s :17.78 kN/m2 r

660 kNII+

1 ' .

1 . 5 r nt , -

I :

253

tI

3 n r

II

A . i

l* t,_s *r

l + : r n + l

Foot ing s izeJ r n x 3 r l

Figure 9.22 Cross section and plan of a column footins

T A

t_,lnfluence value = 0.005

Figure 9.23 Determination of stress at a point by use of Newmark's influence charl

Chapter 9 Sfresses in a Soil Mass

9,10 Summary and General Comments

This chapter presents the relationships for determining vertical stress at a point dueto the application of various types of loading on the surface of a soil mass. The typesof loading considered here are point, l ine, strip, embankment, circular, and rectan-gular. These relationships are derived by integration of Boussinesq's equation for apoint load.

The equations and graphs presented in this chapter are based entirely on theprinciples of the theory of elasticity; however, one must realize the limitations ofthese theories when they are applied to a soil medium. This is because soil deposits,in general, are not homogeneous, perfectly elastic, and isotropic. Hence, some devi-ations from the theoretical stress calculations can be expected in the field. Only alimited number of f ield observations are available in the l iterature at the present t ime.On the basis of these results, it appears that one could expect a difference of *25 to30% between theoretical estimates and actual f ield values.

Prohlems9.1-9.5 For the soil elements shown in Figures 9.24-28, determine the maxi-

mum and minimum principal stresses. Also determine the normal and shearstresses on plane AB. lNote: For Problems 9.1 and 9.2 use Eqs. (9.3), (9.a),(9.6) , and (9.7) ; for Problems 9.3,9.4, and 9.5 use Mohr 's c i rc le. l

9.6 Point loads having magnitudes of 1-5 kN, 20 kN, and 30 kN act at A, B, andC, respectively (Figure 9.29).Determine the increase in vertical stress belowpoint D at a depth of -5 m.

9.7 Refer to Figure 9.30. Determine the stress increase, Aa,, at A, given the fol-lowing data:

4 r : 7 5 k N / m x 1 : 2 m z : 1 . 5 m

Q z : 0 x2 : lm9.8 Repeat Problem 9.7 with the following values:

: 0 x 1 - 5 f t z : 5 f t: 300 lb/ft x2 - 3 ft

100 kN/rn2 400lblfr2

Q r

Qz

-{-0, u*,.. 300rb/ft, + B

45 kN/rn2

60 kN/m2 750 lb/fr2

300 lb/frz

B

Figure 9.24 Soil element for Problem 9.1 Figure 9.25 Soil elememt for Problem 9.2

Problems

80 kN/rnl

--j1,not,.'55 1b/ft2

25 tb/fl

+B

B l

t- -r00 lblfr2

.55 tb/ft2

30 kN/

tm 2

150 kN/m2

*+-I

9.26 Soil element fbr problem 9.3

+*I

9.27 Soil element for problem 9.4Figure

25 kN/m2

-J*,, o*,n,,

I I t 5 k N / r n 2

| , i ' l 'r rrvr' 'I -'-''1 |l,-/+r to" I

fFigure 9.28 Soil element fbr problem 9.-5 Figure 9.29

Line load = r / r L ine load = 4.

-'- l

Ao., I

,l_LF-'',-i

l . + 3 m - l

Figure 9.30

256 Chapter 9 Sfresses in a Soil Mass

I

:- i_-]I '

5 m

II

I Unit weight Y=

Y

II

I

t..t . .| , .t .t * _ _ . ,

1

18 kN/m3

; 1 . : : _ , i - . : ; t i _ i _ : _ ; .

l 1 ' : - ' ; , : _ ' . r _ _ _ " . : , - - - ; : . . ,

; i '

Figure 9.31

9.9 Repeat Problem 9.7 with the l 'ollowing valucs:r 7 1 - l ( X ) k N / m . r 1 - 3 m z . - 2 mq2 - 200 kN/m 'Y. : 2 m

9.10 Rcfer to Figurc 9.6. The magnitude ol thc l inc krad q is 2-500 lb/ft. Calculate

and p lot thc var iat ion o l ' thc vcr t ica l s t rcss increase Arr- between the l imi ts

o f r - l 0 l ' t a n d , r : * l 0 f t . g i v e n t h a t z : 5 f t .9.l l Refer to Figurc 9.30. Given that r71 - l0 kN/m. f,r - 3 fl, rz : 2 m, and z :

1 m, i f the ver t ica l s t ress incrcasc at point ,4 due to the loading is 3 kN/mr.

dctcrmine the magnitude of r72.9.72 Refer to Figure 9.tt. Given that B - 12 tt, r/ : 350 lb l[ l ' , x : 9 ft, and z :

5 f t . determine the ver t ica l s t ress incrcasc, Ao, , a t point .4.

9.13 Repeat Problem 9.12 us ing the fo l lowing values: q : 7000 kN /m' , B : 2 m,

x : 2 m . a n d z - 2 . - 5 m .9.14 An ear th e mbankment d iagram is shown in F igure 9.3 l . Determine the

stress incrcase at point A due to the embankment load.9.15 Figure 9.32 shows an embankment load for a silty clay soil layer. Determine

the vertical strcss increase at points A, B, and C.9.16 Consider a circularly loaded flexible area on the ground surface. Given that

the radius of the circular area is (R) : 4 m and the uniformly distributedload is q : 200 kN/m2, calculate the vertical stress increase Ao. at a point

located 5 m (z) below the ground surface (immediately below the center of

the circular area).9.17 Consider a circularly loaded flexible area on the ground surface. Given that

the radius of the circular area (R) : 6 ft and that the uniformly distributed

load (q) : 4200Ib/ft2, calculate the vertical stress increase Ao. at points 1.5,

3,6,9, and 72 ft below the ground surface (immediately below the center of

the circular area).

e i

f ':i- 5 m

I

Problems 257

7.5 fr 7.5 ftl.el..e

I. t I

- - - - l

z

Unir weight y= I l5 lb/frl

. . , ' : , ' . . " ' . . : . . , ,TI

Figure 9.32

9.18 Figure 9.1-5 shows a l lex ib lc c i rcu lar area o l rac l ius 1? :4 m. Thc uni formlvdis t r ibutcct load on thc c i rcu lar area is 300 kN/mr. Calculatc t l rc vcr t ic t r ls t rcss incrcase i t t r - 0 .0.1J. 1.6.4.6, and lJ r .n . ancl z : 4 . [J nt .

9.19 Rc|er to F igurc 9.33. Thc c i rcu lar l lex ib lc arca is uni l i r rmly loacled. Givenq :320 kN/mr ancl us ing Ncwntark ' .s char t , c lc terrn inc t l . rc vcr t ica l s t ress in-cre i lsc Arr- at point ,4.

9.20 Thc p lan o l 'a l lcx ib lc rcctangular loaclcc l arca is shown in F igurc 9.34. ' l 'he

uni fornt ly d is t r ibu lcc l loacl on thc f lcx ib lc arca. q is 90 kN/mr. I )etcrnt ine thcver t ica l s t rcss incrcasc. Arr - , a t a dctr th o l ' ; : 2 nt bc lowa. Point ,rlb . Point Rc. Point Cl

l = - 4 m - l

r / = 90 kN/rnl

i<- 1.6 nr--- t lJ

0. l i rn

A4 ( '

l+2 r t t+1

Figure 9.34

Cnts.s section

Figure 9.33

1I

30 ft

II: ' t

l 5

+

l rIL - _ _

2

tI

2 n r

IIv - - {

r lao-: I

I o,'af_.L

Chapter 9 Sfresses in a Soil Mass

9.2I Repeat Problcm 9.20. Use Newmark's influence chart for vertical pressured i s t r i hu t i on .

9.22 Refer to the uniformly loaded rectangular area shown in Figure 9.34.Estimate the strcss below the center of the area at a depth of 3.5 m. UseEq. (e.33).

ReferencesAstr,rrN. R. G.. and Ur-r.nv. H. H. (1962). "Tabulated Values for Determining the Complete

Pattern of Stresses. Strains. and Deflect ions Beneath a Uniform Circular Load on a Ho-mogcneous Half Space." in tlighwuy Raseurch Bulletin -?.12, Transportation RescarchBoard . Nat iona l Research C lounc i l . Wash ing ton , D.C. , l -13 .

BcrussrNr,se,J. (18U3). Applicution r les Potentiuls t i L'Etrule de L'Equil ibre et t lu Mouvementdcs So I ida.s E las I iq tras, Gauthier-Vi l lars. Parts.

Da.s. B. (1997). Advunccd Soil Mcclmni<,s, 2nd ed.. Taylor and Francis. Washington, D.C.N cwr. l ,r , r<r. N. M. ( I 942). " lnf lucncc ( lharts for Corlputat ion of Stresses in E,last ic Soi l ." Uni-

vcrsity of l l l inois Enginccring E,xperirnent Stat ion, Bul let in Nr., . - l -18.Os'r 'nnnnn<;, J. O. (1957). " lnf lucnce Valucs for Vert ical Stresses in Semi-lnl lni tc Mass Due

to Embankment Loading," Proct 'cdings, Fourth International Conf'ercncc on Soil Me-chanics and Foundation E,ngineering, I-ondon, Vol. l , 393-396.