bowles on strip loads

8/10/2019 Bowles on Strip Loads

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Figure 11-16 Identification of terms used in the Boussin esq equation [Eq. (11-20)] for lateral pressu re.

Equation (11-20) can also be written as

P \3rh 1 - 2 / x 1

ar

=

2^ [ ^ - - R (RT7) \

( U

-

2

a )

where the several terms including 0, z, r, andR are identified on Fig. 11-16. This form of the

equation is particularly suitable for prog ram min g on a small calculator, since the point load

P

is usually fixed with given

x, y

coordinates and we want to vary

z

to obtain the wall pressure

profile.

The computer programming of this equation allows one to solve any of the given backfill

surcharge loads of Fig. 11-17 defined as follows:

1. Point load. Use the equation in the given form.

2.

Line load. Treat as either one load or a series of concentrated loads along a line of uni

width acting on unit areas.

3 . Strip load. Treat as a series of parallel line loads acting on strips of some un it wid th.

4 . Loaded area. Treat as a series of parallel line loads acting on strips of finite leng th.

We can easily analyze a constant uniformly loaded area (say, the interior part of an em-

bankment) or one with a linear varying load (say, the embankment side slopes). In either of

these cases the loaded area is divided into strips with some load intensity

q

and some small

unit width B, on the order of 0.25 to 0.5 m. Th ese strips are then subd ivided into unit

areas of some lengthL also on the order of 0.25 to 0.50 m. The se un it areas are treated a

a series of point loads of

Q qBL

acting at the center of each of the unit areas. The severa

unit area contributions mak ing up the total loaded area are then sum m ed to obtain the tota

lateral pressure acting at some point at the depth of interest (either in the soil or on a wall)

This is the procedure used in program SM BL Pl (B-8) on your program diskette.

The general validity of using a form of Eq. (11-20) for surcharges was established in sev-

eral publications, including the work of Spangler (193 6), Spangler and Mickle (1956), Re hn-

man and Broms (1972), and others.


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Figure 11-17

Surcharge loads that can be used with the computer program SMBLPl (B-8) on your program

diskette. NSQW, NSQL = number of unit elements away from wall and parallel to wall, respectively, as used in

the computer program.

The early work of Spangler and Spangler and Mickle introduced an error into the general

application of the equations; however, that can be avoided by direct use of Eq. (11-20) and

an appropriate value for Poisson's ratio

/ J L .

When the work of Spangler was first published, he used /J L = 0.5 [and later in Span-

gler and Mickle (1956)], which substantially simplifies Eq. (11-20)but may not be correct

Spangler's work consisted of trying to measure the lateral pressure against a 1.829 m (2.134

m total height) high X 4.572 m wide retaining wall with a constant stem thickness of 0.150

m. He used metal ribbons (since earth-pressure cells were not readily available in the early

1930s) and simply dumped a granular

(WL =

17.5,

Wp =

13.2%) backfill behind the wall

with no compaction at all to produce an extremely loose state. After a time, he had a truck

backed onto the loose backfill so that the rear wheels could simulate two concentrated loads.

To simulate a line load he laid a railroad crosstie parallel to the wall, onto which the rear

wheels (a single axle with dual tires) of a loaded truck were backed. Since the wall was only

4.572 m long and a railroad crosstie is about 3 m long, a strip model was not very likely to

have been produced.

From these efforts Sp angler (both references) found that the meas ured lateral pressure w as

about twice that predicted by Eq. (11-20) with /J L

=

0.5. From the reported results, Mindlin

(1936a), in discussing Spangler's (1936) work suggested that the factor of 2 could be ex-

plained by a rigid wall producing the effect of a mirror load placed sym me trically in front of

the wall (Fig. 11-ISa) . The author began looking at this problem more closely and decided

that the mirror load is not an explanation. As shown in Fig. 11-18&, a mirror load on a rigid

a) Concentrated load

NSQW = NSQL - 1

b)

Line load

NSQW = 1

NSQL = 1 (or as req'd)

(c) Strip load

NSQW =

sufficient for width

NSQL = l (or as req'd)

d) Linear varying load.

e)

Linear varying load.

/ )

Foot ing

load oflimited

BxL.


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(a) Mirrored loads on rigid wall.

Figure 11-18 Th e case for lateral pressu re on a rigid wall.

wall would simply cancel the lateral shear stresses in the wall and certainly not double the

horizontal pressure. A flexible wall could possibly double the lateral pressure but would have

to be extrem ely flexible (and hav e the loads a pplied sequ entially). Re ferring to Fig. 11-18Z?,

we see that the horizontal pressure ah produced by P

0

(applied first) acting on a rigid area of

dydz

would develop shear resistance

r

such that

rt(2dz+2dy) = a^

W hen mirror load

P

0

is

applied, a second shea r stress r w ould deve lop on the elem ent but in the opposite direction, so

the shear stress would cancel and we would simply have on each side of the wall a horizontal

stress ah (not2ah). If the wall is flexible and little shear stress develops, the element would

(if the loads were applied sequentially) displace laterally toward the

-a

direction to prod uce

a resisting soil stress on the

a

side of

a

h

= ah r(2dz

+

2dy).

Since this would become

locked in when P

o

is applied, the stress on the left would become ah + a

h

, and since a

h

is

transmitted through the wall, the right side would also have the existing cr^+ transmitted a

h

value from load

P

0

.

If the loads were applied simultaneously the stresses would simply be

ah

on each side (and not

2ah).

Mindlin got around this complication by inserting a statement that the wall was rigid but

could not carry shear. There is no such wall type known to the author.

Because the Spangler work was done in the early 1930s, it is difficult to speculate on

the cause of the high stresses except to note that the wall had rather finite dimensions. The

surcharge load w as caused b y a truck backing onto the backfill. Wh en it stopped at the desired

position it would have produced an inertial force that was amplified because the fill was not

well compacted. The backfill probably was of limited extent, so that it is also possible some

type of arching occurred that increased the lateral pressure.

More recently Rehnman and Broms (1972) showed (using modern earth-pressure cells)

that when the soil behind the wall was dense the lateral pressure from point loads was much

less than when the soil was loose. They also found that gravelly backfill produced larger

lateral pressures than finer-grained materials. This observation indicates that both soil state

and Poisson's ratio are significant parameters.

Rigid wall

(b) Horizontal stresso

h

on differential

aready dz.


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3 . Con centrated loads well outside the Ra nkin e zone contribute toP

a

in the trial wedge case

leading one to the opinion that the trial wedge is not correct, but conservative.

4 . Since Eq . (11-20) gives small lateral pressures w hen the load is very close to the wa ll, thi

result may mean either that the surcharge load is being carried downward by vertical wall

friction rather than by lateral pressure or that Eq. (11-20) is not valid for a load close to

such a massive discontinuity in the elastic half-space.

5 . Wall pressure s comp uted by Eq. (11-20) are rather small once the wall-to-load d istance i

greater than the Rank ine zone.

Computer Program for Lateral Pressure

Computer program SMBLPl (B-8) on your program diskette can be used for all the lateral

pressure problems shown on Fig. 11-17. By superposition, almost any conceivable surcharge

load can be analyzed, quite rapidly in most cases.

From above table at 1.3 m from wall:

Trial wedge = 65 kN

Boussine sq JR+ Coulomb P

a

gives

Coulomb:

Total

Inside

Outside

TABLE 11-7

Com parison of trial wedge and Bou ssinesq wall forces compu ted using Eq. 11-20).

Also shown is the Coulomb active pressure force.

2.217 m

Load position

from wall, m

Trial wedge, kN

0 0.3

65.0

0.6

65.0

1.0

65.0

1.3

65.0

1.6

64.8

2.0

63.4

2.3

61.6

2.6

59.3

3.0 m

55.5

CoulombP

a

= 17.30) 4)

2

0.276) = 38.2 kN vs. 38.1 of trial wedge)

Boussinesq

fi =

0.3

0.5

0.7

1.0

8.8

13.1

18.0

25.3

8.3

11.5

14.8

19.9

5.3

7.5

9.7

13.0

Inside

4.1

5.8

7.5

10.0

3.3

4.6

6.0

8.1

2.5

3.6

4.6

6.2

Outside

2.1

3.0

3.9

5.2

1.8

2.5

3.3

4.5

1.4

2.1

2.7

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bowles on strip loads

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