J. Hessner

AKADEMIET FOR DE TEKNISKE VIDENSKABER

GEOTEKNISK INSTITUT THE DANISH GEOTECHNICAL INSTITUTE

BULLETIN No. 12

J. BRINCH HANSEN

THE ULTIMATE RESISTANCE OF RIGID PILES

AGAINST TRANSVERSAL FORCES

N. H. CHRISTENSEN

MODEL TESTS WITH TRANSVERSALLY LOADED

RIGID PILES IN SAND

COPENHAGEN 1961

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AKADEMIET FOR DE TEKNISKE VIDENSKABER

GEOTEKNISK INSTITUT THE DANISH GEOTECHNICAL INSTITUTE

BULLETIN No. 12

J. BRINCH HANSEN

THE ULTIMATE RESISTANCE OF RIGID PILES

AGAINST TRANSVERSAL FORCES

N. H. CHRISTENSEN

MODEL TESTS WITH TRANSVERSALLY LOADED

RIGID PILES IN SAND

COPENHAGEN 1961

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S. L. MØLLERS BOGTRYKKERI, KØBENHAVN

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Table of Contents

The Ultimate Resistance of Rigid Piles against Transversal Forces 5 by J. Brinch Hansen, Professor, dr. techn., D. G. I.

1. Introduction 5 2. Pressure at Ground Surface 6 3. Pressure at Moderate Depth 6 4. Pressure at Great Depth 7 5. Pressure at Arbitrary Depth 7 6. Calculation of a Pile 8 7. Example 9

Model Tests with Transversally Loaded Rigid Piles in Sand 10 by N. H. Christensen, Civil Engineer, D. G. I.

1. Introduction 10 2. Extent of Test Series 10 3. Testing Technique , 11 4. Direct Test Results 12 5. Computation of Theoretical Values 14 6. Comparison between Theory and Experiment 14 7. Summary 16

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THE ULTIMATE RESISTANCE OF RIGID PILES AGAINST TRANSVERSAL FOCRES

by J. Brinch Hansen, Professor, dr. techn., DCI.

1. Introduction As shown in fig. 1 we consider a vertical pile with a

cross section B X L and a driving depth D m . It is subjected to a horizontal force H (perpendicular to the sides with width B) acting at a height A above the ground surface.

The ground surface may be loaded with a surcharge p (corresponding f. inst. to the effective weight of a

p I j imi i i i -

C(f

i

A

lUJi <^V/W

Dm

•

JJJJJ, y/Xiy/y

BeD

J tvA

D

JJJ

Ds

J l ^ \

D,

mud layer). The soil proper has a cohesion c and a friction angle <p. The effective unit weight is y above the ground water table and / below.

At an arbitrary depth D below the ground surface the effective overburden pressure is (see fig. 1):

q = p + y D d + y ' D s (1) We assume now that the pile is made so strong that

no yield hinge can develop in it. In the state of failure it can therefore - elastic deformations being disregarded in this connection - be assumed to rotate as a rigid body about a point at a depth D r below the ground surface.

Above the rotation centre passive earth pressures will act on the right side of the pile and active on the left. Below the rotation centre the situation is reversed.

For the resultant (passive minus active) pressure per unit front area of the pile at the depth D we can write the general expression:

eD = q K ^ + ; c K ? (2)

The corresponding pressure per unit length of the pile in fig. 1 is BeD . For a cross section of another shape an "equivalent" width B must be estimated or - better - fixed by means of comparative model tests. In the case of a group of parallel piles (fig. 2) B is to be interpreted as the width of the group perpendi­cular to the direction of the force H.

Pi o c

i

J \H (

J ] (

3 3

s "\

H

Fig. 1. Sketch of pile. Fig. 2. Group of parallel piles.

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2. Pressure at ground surface

For D = 0 we must have ordinary passive and ac­tive earth pressures on the pile, corresponding to the usual plane case:

e0 = q K ° + cK2 (3)

For K° we insert here the difference between the passive and active coefficients, corresponding to. a rough wall which is being translated horizontally [1 , 2 , 3 } :

K° = e(* n + V) tan <p c o s g, t a n (45° + £ ^

_ e - ( | t t - ( p ) t a n < p C O S ( p t a n ( 4 5 o _ . l ( ^ (4)

As regards K° we disregard - on the safe side - the corresponding active pressure-term, because it might lead to negative earth pressures on the active side of the pile. For the passive pressure-term alone we get:

Kc0 = [ e ( i " + <P) tan<Pcosg?tan {45°+i<p) - l ] c o t y (5)

Curves for K° and K° are indicated in fig. 3.

800 | | | ! | I I | | | | | | I | | | | i | 1 M 1 1 1 1 1 1 1 1 1 1 M 1 1 1 1 1 1 1; finn K F

000 1 1 Li

1 1 1 1 1 1 / /

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11 i IM11MI i 1111111 i 11 i 11 II II I II II Li 11 II I W i " M M

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M -HT q/l UT i- U-Hil M J-Hi 12 \ P n \ r l UKKI

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c K I LVmTH^r - mllm J [ 'fJ-rrn J-T 1 Ka M \l\\jnf\T LUil in . J I M IJJ-TTIT 1 iA\ 1111111 Lrll 1111II11111111111 • iWrttTn ml MN

-oH Kl JUll / m l ir m L-rTM Lrl rn 1 ' I 17 1 1 1 1 1 1 1 / I 1 1 1 K I 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Cf)

;:;[llLLLi^Mlllllllllllllllllllllllllllll(P, - 0° 5° 10° 15° 20° 25° 30° 35° 40° « 0

, Fig. 3- Earth pressure coefficients at ground surface (0) and at greath depth ( oo ) .

3. Pressure at moderate depth

In order to find the increase of the earth pressure with depth (for reasonably small depths) we consider the simplest possible case, viz. the passive Rankine state (fig. 4 ) . . •

Fig. 4. Earth wedge in the Rankine case.

We assume here that the earth wedge is bounded by two vertical planes (distance B) , and that in these

planes shearing forces develop, corresponding to a nor­mal pressure equal to the earth pressure at rest ( qK 0 ) . The shearing stress at a depth D below the surface will then be:

rf = c + ( y D + p ) K 0 tan 99 (6)

We consider now a strip of earth with a width B and a height dD, bounded by two failure planes (fig. 4) . We assume that the direction of the failure planes is the same as in the plane case; this is sufficient­ly correct for reasonably small depths. Projecting all acting forces on a plane making the angle cp with the failure plane we get then an equation, which can be brought on the following form:

e r = p t a n 2 ( 4 5 0 + i y )

+ 7Dtan2(45° + 4-g))

+f D

2 Ko sin q, sin (45° + i<p) J

Ko sin q> sin (45° + i cp) i

+ 2 c t a n ( 4 5 0 + i g P ) . [ l + ^ ' . ' 2 sin (45° + ^ ) J (7)

This equation has been developed for a smooth wall, but as an approximation we shall use the factors in the

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brackets for our rough wall too, and apply them to the resultant (passive minus active) pressures. Moreover, as the surface load p is usually small, we shall for the sake of simplicity - and on the safe side - use the y-factor for p instead of the p-factor. Thus we get:

e = qK;

D

Ko sin q> sin ( 4 5 ° + 105) J

+ cK° 1 + ^ - 2 sin (45° + ^cp) (8)

4. Pressure at great depth

At great depths the rupture-lines will not go up to the ground surface any more, but will instead go horizontally around the pile [2, 3, 4}. We have then again a plane case of failure, but the planes are now horizontal.

In order to calculate the corresponding passive pres­sure on the pile we shall consider the somewhat similar case of a deep strip foundation. In a recent paper [5] the author has proposed the following formulas for this case:

b = i y B N y d y + ( c + q t a n y ) N c d c - f q (9) B , 0.6 1 ( 1 0 ) de = 1 + 0 . 3 5 : ^ + ^

' tan4g5J

If these formulas shall be applied to the passive pressure on our pile at great depths, the following changes must be made. The first and the last term in (9) are left out, and the effective vertical pressure (q)

in the second term is substituted by the horizontal pres­sure at rest (qK0 ). Finally, in (10) we can put D = oo. Thus we get:

e00 = (c + qKo tan y) Ned? = q K £ +cK cæ (11)

d ' = 1.58 +4.09 tan* q> (12) Nc is the usual bearing capacity factor, for which

the following formula is valid [2, 3, 5]:

Nc =[e 3 i : t a n9 , tana(45 0+ig))- l ]cotgr) (13) For K0 the following formula has been proposed

[6}: Ko = 1 -sin cp (14)

The constants K" and Kq can now be calculated: K ^ N c d ? (15) K£ = K? • K0 tan p = Ncd?K0 tan <p (16)

Curves for K" and K" are indicated in fig. 3.

5. Pressure at arbitrary depth

We need now two general formulas for the con­stants K° and Kf to be inserted in equation (2) in

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order to give us the pressure eD at an arbitrary depth D.

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D Lt 11

759

272

118

61.4

36.8

24.5

17.6 112 102 8.14

0 5 10 15

Fig. 5. Earth pressure coefficient for overburden 20

pressure.

5 10 15 20

Fig. 6. Earth pressure coefficient for cohesion.

w w w • g e o • d k

For D -*- 0 we must require that K D - ^ K0, and for D —- oo that KD — K00. Also, for moderate depths, the increase of the constants with depth should be expressed by the brackets in equation (8). The simplest empirical formulas, which fullfil all the above re­quirements, are the following:

(17) K? =

" q —

K q + Kq

l + aq

K0q

Kq - Kq

- •§ D B

KoS

sin (45° in go

+ i 9) (18)

K?: Kj + Kc" -etc B

O r =

l + ac

K°

I) B

2 sin (45° + ^ cp)

(19)

(20) Kr-K°

In fig. 5 and 6 are given curves for K° and Kf re­spectively as functions of cp and D : B. These curves, together with the simple formula (2), enable a direct and quick determination of the horizontal pressures on the pile. The values at the arrows (right) correspond to D = oo.

6. Calculation of a pile

Usually the height A of the force H above ground level will be given. If the driving depth Dm is also given, the two unknown quantities are the depth Dr of the rotation centre and the ultimate value of the force H. They are determined by means of the two equili­brium conditions (horizontal projection and moment equation). First, D, is fixed by trial in such a way, that the two pressure areas will give equal moments about the line of the force H. Then, this force is found as the difference between the two pressure areas (fig. 7).

When the pile should be designed to resist a given force H, the unknown quantities are the driving depth Dm and the depth D r of the rotation centre. B must be estimated in advance. The necessary safety can be introduced in the design either by multiplying the force H by a "total" safety factor F, or by applying "partial" coefficients of safety to H, q? and c [2, 3, 7, 8}. In the latter case the calculation is made with the "nominal" quantities:

Hn = H . fH = H - 1.5 (21) t a n c p t z n c p

tangpn =

c„ = ^ =

1.2

Tc-r5 W The calculation now proceeds in the following way.

First, the depth D0, at which the transversal force is zero, i. e. where the maximum moment occurs, is de­termined by the condition that the pressure area above this point should be equal to Hn. The moment Mn at this point is then calculated, and for this the cross section of the pile should be designed with nominal stresses in the material. Finally, the driving depth and the rotation centre are fixed - by trial - in such a way, that the two additional pressure areas are numerically equal and give a moment equal to Mn.

If we have layered earth with different values of y, c and cp in the different layers, q is always found from (1), but in each layer the pertaining values of c and cp are used for the calculation of the pressures in this layer.

When the effective shear strength parameters c and cp are used in the calculation, this corresponds, to the long-term resistance. For the determination of the short-term resistance, which will develop under im­pact forces, the undrained parameters qi — 0° and c = cu must be used in fully saturated layers of clay or silt. In sand layers the excess pore pressures dissipate usu­ally so quickly, that - even for impact forces - only the ordinary (long-term) parameters should be con­sidered [2, 3, 9] .

Fig. 7. Earth pressure diagram for pile.

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7. Example

We shall here consider the following example of a pile in gravel (fig. 8 ) , which has been treated pre­viously by H. Blum [10}:

A = 5.0 m B = 1 . 3 m D m ; = 5 . 4 m / = 1.0t/m3 cp = 35° c = 0

For cp = 35° we find by means of fig. 5 the follow­ing results at 6 equally spaced points of the driving depth:

0 1.08 2.16 0 0.83 1.66 7.0 8.6 9.9 0 12 28

The corresponding pressure diagram is shown in fig. 8. We try first the lowest fifth point as a possible rotation centre. The moment equation about the force line gives then: A M = 12 • 1.08 • 6.08 + 28 • 1.08 • 7.16

+ 47 • 1.08 • 8.24 + 69 • 0.54 • 8.96 - 69 • 0.54 • 9.68 - 93 • 0.54 • 10.04 = 79 + 217 + 418 + 334 - 36l - 504 = 1048 - 865 = 183 tm

The rotation centre should then be raised about: 183

-~*H=24.0t

D D : B

Kq

/ D E K ,

3.24 2.49

11.2 47

4.32 3.32

12.3 69

5.40 4.15

13.3 93

Ah = = 0.14m 2 • 69 • 9.32

At this point we find D = 4.18 m, D : B = 3.22, Kq. = 12.2 and / DBKq = 66 t/m2. Consequently, a more accurate value of Ah is:

183 Ah = = 0.147 m

2 • 67.5 • 9-25 The ultimate value of the force H is then:

H = 12 • 1.08 + 28 • 1.08 + 47 • 1.08 - 9 3 - 0 . 5 4 - 2 - 6 7 . 5 - 0 . 1 4 7 = 13.0 + 30.2 + 50.8 - 50.2 - 19.8 = 24.0 /

For comparison it can be mentioned that Blum finds H = 27 t in this case.

Fig. 8. Example of pile calculation.

References. [1]. J. Brinch Hansen: Earth pressure calculation. Teknisk Forlag, Copenhagen 1953. [2]. H. Lundgren and J. Brinch Hansen: Geoteknik. Teknisk Forlag, Copenhagen 1958. [3]. J. Brinch Hansen and H. Lundgren: Hauptprobleme der Bodenmechanik. Springer-Verlag,

Berlin I960. [4]. J. Brinch Hansen: The stabilizing effect of piles in clay. CN-Post No. 3, Nov. 1948. [5]. J. Brinch Hansen: A general formula for bearing capacity. Ingeniøren, International Edi­

tion, June 1961 and Geoteknisk Institut, Bulletin No. 11, Copenhagen 1961. [6]. A. W. Bishop: Test requirements for measuring the coefficient of earth pressure at rest. Proc.

Conf. Earth Pressures, Vol. I, Brussels 1958. [7]. J. Brinch Hansen: Brudstadieberegning og partialsikkerheder i geoteknikken. Ingeniøren, 11-5-

1956 and Geoteknisk Institut, Bulletin No. 1, Copenhagen 1956. [8]. J. Brinch Hansen: Definition und Grosse des Sicherheitsgrades im Erd- und Grundbau. Bau­

ingenieur 1959, Heft 3. [9]. J. Brinch Hansen: Om jordarternes forskydningsstyrke. Korttids- og langtidsstabilitet. Inge­

niøren, 15-6-1958 and Geoteknisk Institut, Bulletin No. 3, Copenhagen 1958. [10]. H. Blum: Wirtschaftliche Dalbenformen und deren Berechnung. Bautechnik 1932, Heft 5.

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MODEL TESTS WITH TRANSVERSALLY LOADED RIGID PILES IN SAND

by N . H. Christensen, Civil Engineer, D G I

1. Introduction The present paper deals with a series of tests car­

ried out on wooden model piles in sand to determine their resistance against transversal forces. The ultimate transversal load was determined for different pile depths, heights of load application, densities of sand, and methods of placing the pile.

The purpose of the tests was to investigate the vali­dity of the theory set forward by Professor Brinch Hansen in the preceding paper, which gives the trans­versal failure load on a pile as a function of the pile geometry and the friction angle of the sand.

2. Extent of Test Series

The total number of tests was 26. All piles had a square cross section of 5 cm X 5 cm. The parameters varying from test to test were the following (see fig. 1):

The depth D of the pile below the horizontal sand surface was either D = 25 cm (symbol s for short pile) or D = 50 cm (symbol / for long pile).

The height A of the horizontal load above the sand

S.

5hr

^

'•: •.'lw: • ' . : : • v . ••''„J"- •;

U

M

A=22.5

0-25 •*

sh

All dimensions are cm

surface was either A = 22.5 cm (symbol Å for higher force line) or A = 5 cm (symbol / for lower force line).

The pile was either driven into the sand with an ordinary hammer (symbol^) or was "preplaced" (sym­bol / ' ) , i. e. placed in position before the sand was fil­led into the box.

-UH £X • . . i .

D=25 '

Si

r ^

A=22.5

D=50 . *

ki D-50

lh It ?ig. 1. Position of dial gauges on piles. Pile dimensions and height of transversal force in the four test types.

10

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The sand was either loosely deposited or densely packed in the test box.

After tests had been made for a number of the 16 possible combinations, it proved necessary to modify the definition of failure, and for this reason 8 tests were doubled. However, it was found later that for such of the first tests which had not reached failure according to the new definition, a relatively accurate

failure load could yet be found by extrapolation. Final­ly, one more test was repeated twice because of its sur­prisingly great ultimate load; the results of the two dup­licate tests (No. 25 and 26) turned out to be more con­sistent with the other tests. On the basis of this the original test (No. 3) , the result of which is shown in fig. 9, is disregarded in the final conclusions.

3. Testing Technique

Apart from the placing of the sand and the piles in the box, the tests were all carried out in the same way.

3.1 The Box and the Sand.

The sand box had a base of 1 m X 1 m and a height of 0.7 m. With this size it was possible, with due re­gard to the extension of the ruptures in the sand, to conduct four pile tests for each time the box was filled up.

The sand was the so-called 'Gl2-sand' which is used for most experiments on sand in this laboratory; its pro­perties are therefore well known, and a homogenous test medium is ensured.

A grain-size distribution curve for the Gl2-sand is shown in fig. 2.

As already mentioned, two different densities were used. When the sand was deposited by means of a sieve, the void ratio e in the box would assume values from 0.64 to 0.71. Void ratios between 0.53 and 0.56 were obtained when the sand after sieving was com­pacted thoroughly in layers of 10 cm thickness.

3.2 The PUes.

In fig. 1 are shown the different pile dimensions used. The piles were made of wood, and the part below

1UU7.

807.

607.

407.

207,

07. _ ^ . J

f I

\

0.02 Q06 a i 0.2 0.6 1.0

Fig. 2. Grain-size distribution of the G 12-sand.

zo mm

the sand surface was covered with sandpaper to ensure complete roughness. Eye bolts were screwed in at the points of application of the loading force.

3.3 The Gauges. The measurements taken during the tests comprised

the horizontal deflections at two points of the pile, the vertical deflection of the pile top, and the horizontal force acting on the pile. The deflections were measured by three ordinary dial gauges, and the horizontal force by means of a proving ring provided with a dial gauge.

The position of the gauges is shown in fig. 1.

3.4 Running the Test. It will be seen from fig. 1 that the two employed

lines of action of the transversal force on the pile were the same lines as those in which the horizontal deflec­tions were measured.

The horizontal load on the pile was provided by an electromotor working through a gearbox, a spindle, and the above-mentioned proving ring. This arrange­ment ensured a suitably low and steady deformation rate, varying in the different tests from 5 to 15 cm per hour in the force line.

While the pile was moving, continuous readings were made of all gauges. This could be done by one man, as the slow deformations permitted all readings in a set of four to be regarded as simultaneous.

In the first tests the pile was vertical at the start, and the test was stopped when an inclination of 1 : 10-1 : 5 had been reached. In the later tests the pile was placed with a negative inclination (backwards) of 1 : 10, and would pass through the vertical position during the test which was stopped at a positive inclination of about 1 : 10.

Furthermore, during the later tests the sand surface was continually readjusted so that it remained plane and horizontal; this was not the case in the first tests. But it seems that this circumstance has had no signifi­cant influence on the test results.

3.5 Void Ratio Determinations.

When filling the box it was not easy, and not neces­sary either, to obtain exactly the void ratio aimed at.

11

w w w • g e o • d k

Roughly speaking, the tests were meant to be conduc­ted at only two different densities, but minor varia­tions in these were, of course, quite acceptable as long as the void ratio e was determined for each test with the best possible accuracy. This was necessary as the ultimate load on the pile increased greatly with the friction angle of the sand.

The void ratio determinations were made by means of an ordinary vacuum-cleaner. A cylindrical tube is pressed down into the sand, and from the interior of this a known volume of intact sand is removed into a bottle for weighing by the use of two mouth-pieces of different lengths. This method has proved very reliable in dry sand, provided that the method is calibrated by performing it on reference sand fillings, obtained by pouring a known weight of sand into a small box with a known volume.

In each filling of the model box there would gene­rally be two piles placed beforehand, and two to be driven into the sand. As the pile driving probably af­fects the void ratio distribution in the box, the tests on the preplaced piles, and the determination of void ra­

tios for these tests, had to be carried out before the driving of the other two piles. These void ratio deter­minations could therefore only be made in the surface layer of the sand.

After the driving and the tests on the driven piles, void ratio determinations were made for these latter tests, and in this case it was possible, and more de­sirable too, to determine void ratios in a deeper layer of the sand as well as at the surface.

Thus for each filling of the box 12 to 15 void ratio determinations were made, about one third distributed in the sand surface around the "preplaced" piles, one third in the surface around the driven piles, and the rest around the driven piles at about mid-depth of the longer piles.

This procedure made it possible to determine a void ratio for each test pile as a weighted mean of a number of void ratios at different distances from the pile, al­though the deviations of void ratios in one filling of the box were generally not so great that any careful estimate of weights was called for.

The value of e found for each test is given in table I.

4. Direct Test Results

10 12 14 kg

23 cm

Fig. 3. Test no. 13 (sip): load-deflection curve, with variation of D r (depth of rotation centre). The deformation tan v is the an­

gular deflection.

During each test between 60 and 80 sets of dial readings were taken. It was found that the rotation centre of the pile usually moved little during the test,

2Acm

Fig. 4. Test no. 25 (r/W): load-deflection curve, with variation

.f O r .

12

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apart from certain deviations at the start. The inclina­tion tanv of the pile relative to its starting position was therefore chosen to represent the deformations; it was determined by the two horizontal deflections measured. The transversal force H on the pile is determined by the deformation of the proving ring. On this basis a load-deflection curve (tanr, H) could be drawn for each test.

The depth Dr of the rotation centre is determined as

Dr = dtanv -.A

where åH is the deflection measured in the force line; a (tanf, Dr)-curve could thus be plotted in the same graph.

In the calculations carried out on the basis of the theory (see section 5) the theoretical value Drt of Dr

at failure was determined for each test, and compared with the experimental values. The experimental values of Dr were most often found to be slightly smaller than the theoretical values. It must be noted, however,

that the experimental determinations of Dr in the manner described are probably less accurate than those of the failure loads H; and Dr may be more influenced by an uneven distribution of e in the box than H is.

Examples of load-deflection curves are shown in figs. 3-6.

Most of the tests showed no definite ultimate value of H; it was therefore necessary in these tests to define the failure load as the load corresponding to a certain deformation. When a number of different test types had been tried out it was found that tan^ = 0.2 was a suitable failure deformation.

In a few of the tests, H reached a maximum for a smaller value of t&nv and then decreased with further deformation (see fig. 3); for these tests the failure load was defined as the maximum value of H. This shape of load-deflection curve was mostly found for small void ratios and short piles.

The failure loads, determined by these definitions, are given in table I.

16 24 32 40 48 56 kg

16 20 24 28 32 36 40 44 cm 30 32 34 36 38 40 42 44 cm

Fig. 5. Test no. 22 ( l id) : load-deflection curve, with variation Fig. 6. Test no. 19 (Ihp): load-deflection curve, with variation of D , . of D , .

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Test No.

1

3 25 26 10 11

2 17 4

18

9 21 12 22

6 8

13 15

5 19

7 20 14

23 16 24

Test Type

shp shd shd shd sip sld Ihp Ihp Ihd Ibd Up Up Ud Ud

shp shd sip sld Ihp Ihp Ihd Ihd Up Up

• Ud Ud

Void ratio

e

0.654 0.635 0.689 0.712 0.695 0.640 0.650 0.675 0.637 0.635 0.693 0.673 0.640 0.626

0.542 0.541 0.557 0.532 0.551 0.548 0.541 0.547 0.558 0.554 0.546 0.553

TABT.F, I.

qjpj

(from fig. 9)

37!8 38°6 36°4 35?5 36?2 38°4

.38°0 37°0

38° 5 38^6 36° 3 37°0 38°4 39! 0

42^8 42^9 42°1

43!3 42^4 42^5 42°9 42°6 42°0 42°2 42^6 42°3

Ultimate load

H(kg)*)

4.18 6.45 3.80 3.80 6.30 8.25

(26.0) 26.8

(34.5) 32.3

(35.0)

39.3 (44.0) 44.5

9.95 10.2 13.8

16.3 (60.0) 50.2

(59.0) 53.5 72.0 70.5 78.3 73.4

By <dm2)

5.21 7.96 4.84

4.91 8.06

10.2 32.4

33.9 42.6

39.9 46.0 49.6 52.0 54.6

11.6

11.9 16.0

18.9 70.2 58.7 68.6 62.5 83.7 82.7 91.0 86.0

f (experimen­tal value)

i l ° l 42°6 36°8

36f9 36^4 39! 3 • 36^4 37!0 39?8 39?0 36°7 37°6 38^2 38!8

4 5 ° ! 45^8 44!o 450.4 44 ! 5 43! l 44!4 43!6 43!3 43!2 44!0 43!5

coiq,

1.295 1.089 1.338

1.331 1.356 1.221 1.356 1.328 1.202 1.236 1.344 1.298 1.270 1.244

0.976 0.971 1.034 0.986 1.016 1.070 1.022

1.049 1.061 1.065 1.035

• 1.053

*) Values in parenthesis were found by extrapolation.

5. Computation of Theoretical Values

The theory set forward by Professor Brinch Hansen in the preceding paper gives the horizontal failure load on a vertical pile in sand as a function of the pile geometry, the angle of internal friction and the unit weight of the sand. The present test series comprises, as regards pile geometry, four different types, viz. sb, si, lh, and // as defined i fig. 1, whereas the variations of the friction angle and the unit weight are more or less irregular.

The computations necessary for an adaptation of the theory to the present experiments consisted of a theo­retical determination of HJBy (where H is the transver­sal failure load, B the width of the pile, and y the unit weight of the sand) as a function of the friction angle

cp, for the four different cases of geometry. These com­putations were carried out for four values of cp: 30°, 35°, 40°, and 45°, and in analogy with the example in the preceding paper.

The results are shown as the curves sh, si, lh, and // in the {cp, HI By) -diagram in fig. 7. Concerning the points indicated, see later.

The theoretical value D r t of D r , the depth of the rotation centre, was determined by the calculations as a function of cp for each test type. D r l differed only little from 0.8 D. The results are shown as the curves in the (9?, D r t )-diagram in fig. 8. The points marked in fig. 8 are the experimental values of D r plotted against the qp-values of the tests as given in fig. 9.

6. Comparison Between Theory and Experiment

The test results as given in table I consist of 26 sets of values of the void ratio e and the failure load H.

The actual friction angle cp of the G 12-sand used in the tests can be found as a function of the void ratio e;

but the relation {e, cp) has proved to be different for different types of tests. As an example the friction angle 9? that determines the active earth pressure on a model wall, in a condition of plane deformations, is

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w w w • g e o • d k

dm

^(\r\

60

cn UU

OU

40

i n

on zu ^c 10

17 1Z

10 o o

c b

0

/ 4

•s J

2S

H BY

/

X

s

s '

y

s -

s

\y Ih /

'

SJ' x

^ ^

^ ^

^ • ^

^

y*

s *

s s *

^

^

V

^

D ^ ir—

S

| X

^

K' ^

* i

^

s *

S

\ | > i

V

>

V i i i i

s

Y s

.

s

s s

1

y

s —̂

y

/

i

^ s

A s

S

S

j \ w A

i r

t — i , ^

i i

y

/

/

X 1

s '

/

/ s

/

J

s

(D H

30° 35° 40° 45° Fig. 7. Relation between friction angle q> and failure load i l , as given by the theory for the four test types sh, si, lh, and //. Experimental results (with symbols as in fig. 9) assuming q, in the present tests to be given by the relation (e, cp) valid for plane

active earth pressure tests.

found [1} to be about 4 0 -5° greater than cp found in triaxial tests [2,3] for the same void ratio (provided that <p is determined by the straight line envelope to the Mohr circles).

On the basis of this knowledge the best way to interprete the results of the present experiments will be to determine the friction angles cp which, according to the theory, correspond to the experimental failure loads, and to compare the qs-values obtained in this way (as a function of e) with the results of other test types, e. g. triaxial and active earth pressure tests.

As e and H are known, H/By is easily computed, and the corresponding "experimental" value of cp is found by means of the curves in fig. 7. The results are given in table I, as well as in fig. 9.

cm

40

30

20

10

n u r

—

1

— a

T D

•

a

+ •

0

•

V

O

X

n_ UT

si

sh o

<>

&

o

A

V

- X

•

<p 35« 40° 45°

Fig. 8. Relation between friction angle q) and depth D r of rotation point as given by the theory for the four test types sh, st, lh, and //. Experimental results (with symbols as in fig. 9) , cp being de­termined (by the curves in fig. 7) as the value that corresponds

theoretically to the failure load in the test.

Fig. 9 represents the main results of the test series. sTo compare with the experimental values {e, cot cp), in­dicated by points, two lines are shown, giving the rela­tion {e, cotcp) as found from triaxial tests and active earth pressure tests on the same sand. Apart from a possible small scale effect, the pile geometry and the manner in which the pile was placed have apparently had no significant influence on the test results as pre­sented in fig. 9. This means that Professor Brinch Han­sen's theory of ultimate horizontal loads on rigid, ver­tical piles is confirmed, in so far as the present test series has yielded a relation {e, cotcp) that conforms well to the one found for the active earth pressure tests on the Gl2-sand. The differences in cp are only 0 o -2 o , and are practically always on the safe side.

The 99^-values corresponding to the void ratio in each pile test, i. e. the values cp = cpti determined by the relation {e, cot 93) found for the active earth pres­sure tests, are given in table I. If cpti is regarded as the 'true' friction angle, another comparison between the present experiments and the theory becomes possible. The experimental results may then be plotted in a (931„ HJBy)-diagram to compare with the theoretical curves; this has been done in fig. 7.

15

w w w • g e o • d k

oshd vlhd •shp Aihp xs ld o l l d +slp a l tp

0.52 0.5/i 0.56 0.58 0.60 0fi2 0.64 0.66 0.68 0.70 0.72 Fig. 9. Correlation of void ratio e and friction angle q> in pile tests, cp being determined (by the curves in fig. 7) as the value that

corresponds theoretically to the failure load in the test.

7. Summary

26 model tests have been carried out on wooden piles in dry sand to determine their resistance against horizontal loads. The parameters varying from test to test were the pile depth, the height of the horizontal load, the density of the sand, and the way the pile was / placed in the sand. The test results were used for draw­ing up load-deflection curves.

Computations were carried out in accordance with the theory set forward in the preceding paper. These were, for each test, a determination of the friction angle that would, according to the theory, give the same ultimate load as found in the test. These friction angles are plotted (fig. 9) against the void ratio of the sand as the main results of the present test series.

It is seen that the "experimental" friction angle cp is significantly influenced only by the void ratio of the sand. The values of 93 are only slightly greater (0o-2o) than those found in active earth pressure tests, and their dependence on the void ratio conforms well to those earlier tests.

The present model tests can therefore be regarded as a reasonable confirmation of the theory, indicating that it is usually a little on the safe side to use this theory, provided that cp is taken as corresponding to a case of plane strain.

Kefe eferences [1]. N . H. Christensen: Model Tests on Plane Active Earth Pressures in Sand. Geoteknik Institut,

Bulletin No. 10, Copenhagen 1961. [2]. Bent Hansen and D. Odgaard: Bearing Capacity Tests on Circular Plates on Sand. Geoteknisk

Institut, Bulletin No. 8, Copenhagen i960. [3]. Bent Hansen: The Beating Capacity of Sand, Tested by Loading Circular Plates. Proc 5th Int

Conf Soil Mech Found Engn, Vol I pp 665-668, Paris 1961.

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