STIFFNESS COEFFICIENTS OF LAYERED SOIL SYSTEMS

By A. Sridharan,1 N. S. V. V. S. J. Gandhi,2 and S. Suresh3

ABSTRACT: One of the most important dynamic properties required in the design of machine foundations is the stiffness or spring constant of the supporting soil. For a layered soil system, the stiffness obtained from an idealization of soils un­derneath as springs in series gives the same value of stiffness regardless of the location and extent of individual soil layers with respect to the base of the foun­dation. This paper aims to develop the importance of the relative positioning of soil layers and their thickness beneath the foundation. A simple and approximate procedure called the weighted average method has been proposed to obtain the equivalent stiffness of a layered soil system knowing the individual values of the layers, their relative position with respect to foundation base, and their thicknesses. The theoretically estimated values from the weighted average method are compared with those obtained by conducting field vibration tests using a square footing over different two- and three-layered systems and are found to be very good. The tests were conducted over a range of static and dynamic loads using three different materials. The results are also compared with the existing methods available in the literature.

INTRODUCTION

Several theories have been proposed to design machine foundations resting on soils that idealize these as homogeneous elastic half-spaces. In practice, however, these assumptions are rarely fulfilled. The most common of the various complexities is the presence of layering. One of the frequently used techniques to analyze the dynamic response of a layered soil system is to replace the layering by an equivalent homogeneous layer with the equivalent parameters. This paper aims to present an approximate method to obtain an estimate of the equivalent spring constant/stiffness of a layered system when the individual spring constants are known.

For foundations resting on layered systems subjected to dynamic loading, the conventional analysis based on the mass-spring-dashpot model is gen­erally recommended. In this idealization, the layered system is represented by a series of springs. The equivalent spring constant or stiffness can be calculated using the following equation. For a layered system having n layers

where keq = equivalent spring constant of layered systems; and fci,2...,„ = spring constants of individual layers.

'Prof., Dept. of Civ. Engrg., Indian Inst, of Sci., Bangalore 560 012, India. 2Asst. Prof., Dept. of Civ. Engrg., JNTU Coll. of Engrg., Kakinada 533 003,

A.P. India. 3Post Grad. Student, Dept. of Civ. Engrg., Indian Inst, of Sci., Bangalore 560

012, India. Note. Discussion open until September 1, 1990. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The •:•• manuscript for this paper was submitted for review and possible publication on Au­

gust 12, 1988. This paper is part of the Journal of GeotechnicalEngineering, Vol. 116, No. 4, April, 1990. ©ASCE, ISSN 0733-9410/90/0004-0604/$1.00 + $.15 per page. Paper No. 24548.

604

Eq. 1 gives the same value of stiffness irrespective of the thickness of individual layers and their relative position with respect to the base of the foundation. This is a serious limitation. This paper represents an initial at­tempt at solving this problem. It will be shown that the thickness and po­sition of the individual layers have a significant influence on the equivalent stiffness.

THEORETICAL CONSIDERATIONS

Reissner (1936) and Nijboer (1956) have shown that expressions obtained using static analysis could be applied to dynamic analysis as well. This has been supported by Heukelom and Foster (1960). As an analogy to these statements, an attempt has been made in this paper to predict the equivalent spring constant of a layered system using static methods.

Palmer and Barber (1940) have given a procedure for obtaining the equiv­alent modulus of elasticity for a two-layer system when the individual thick­nesses and moduli are known. The final expression as given by them could be stated as

E2 E2

Ee E,

1

tf(Eyw 1 + d \E2.

(2)

where Fco = ratio of Young's modulus of bottom layer Ez to that of the equivalent half-space Ee; Eu E2, and Ee = Young's moduli of layers 1, 2, and the equivalent half-space, respectively; h = thickness; and r0 = radius of footing. Thus, knowing the displacement, one can get the equivalent spring constant as a ratio of load to displacement.

For a three-layer system, the upper two layers are replaced by a single layer of thickness (hi + h2), and the equivalent modulus is given by Thenn de Barrows (1966) as

Ee = h ^ + h2-Z/Et\

(3)

Eq. 3 can be rewritten in terms of the stiffnesses of the corresponding in­dividual layers, and the resulting equivalent stiffness can be obtained for a three-layer system.

Odemark's (1949) method for obtaining the displacement factor for a three-layered system is a complicated one. It has been described in detail by Uesh-ita and Meyerhof (1967). The method involves an iterative procedure and a series of computations to obtain Fco. The final equation to obtain the modular ratio E3/Ee is as follows (Ueshita and Meyerhof 1967):

E„

r0.

r0.

1/2

1 + N22

h\ -iV2

>0,

1 -

1 +N22

r0.

(4)

605

1 I 1 J J 0 0-4 0-8

h, Layer 1, k,

h2 Layer 2, k2

3B

h3 Layer 3, k3

h4 Layer 4 , k4

FIG. 1. Layered System Used in Analysis

S l i

FIG. 2. Theoretical Curve Used in Analysis

606

TABLE 1. Scheme of Investigation for Two-Layer Systems

Series number

(D 1 2 3 4 5

6 7 8

9 10 11 12 13 14

Material in top layer (2)

Sawdust

Sand passing No. 7 sieve

Granite chips of size 6 mm

Thickness (cm) (3)

5.7 11.5 25.0 46.0 69.0

25.0 46.0 69.0

5.7 11.5 25.0 34.5 57.5 69.0

TABLE 2. Scheme of Investigation for Three-Layer Systems

Series number (1) 1 2 3 4

5 6 7 8 9

Material (2)

Sawdust

Sand

.ayer 1

Thickness (cm) (3)

23.0 34.5 46.0 69.0

11.5 23.0 34.5 57.5 69.0

Material (4)

Sand

Sawdust

Layer 2

Thickness (cm) (5)

46.0 34.5 23.0 0

57.5 46.0 34.5 11.5 0

The expressions to obtain N2, N2, and N3 involve the ratio E3/Ee and are as follows:

^ (Ex E3\1/3

#2 = 0 . 9 - — - — (5a) h \E3 EJ

h, (EX E3\ h2 N2 = 0.9- ( — • — ) + 0 . 9 - (5b)

h \E3 EJ h

hx (Ex E3\1/3 h2 (E2\

l/3

JV3 = 0 . 9 - —•— + 0 . 9 - — (5c) h \E3 EJ h \E3J

where hx and h2 = thickness of the top and intermediary layers, respectively;

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h = hi + h2; and Eu E2, and E3 = modulus of elasticity of layers 1, 2, and 3, respectively.

Eq. 4 can be used to evaluate Fco for a three-layer system. By equating hx and Ex to zero, the same can be used for a two-layer system as well.

These two methods have been shown to compare well with those of Bur-mister (1943, 1945) by Palmer and Barber (1940) and Ueshita and Meyerhof (1967). In this paper, Odemark's method has been chosen for comparison purposes.

WEIGHTED AVERAGE METHOD

This method employs a simple numerical procedure to determine the equivalent spring constant of a layered system when the spring constants of the individual layers are known. Fig. 1 shows a typical layered system of four layers. In this investigation, the analysis is based on Boussinesq theory

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FIG. 5. Variation of Maximum Displacement Amplitude with Thickness of Sand (as Top Layer) in Two-Layered System

and can be used when any number of layers are present. Akai et al. (1971) have reported some model studies on stress distribution in layered soil sys­tems, in which an attempt has been made to verify the validity of Boussinesq theory for layered systems. It has been shown that the theory is quite ac­curate for layered systems. Thus, in this investigation, it is assumed that Boussinesq theory may be used without significant loss of accuracy. It is also assumed that the layers are perfectly horizontal and that the individual properties do not vary with thickness. Based on the observions of Eastwood (1953) and Arnold et al. (1955), the effective depth of influence is assumed to be three times the width of footing, i.e., within the depth of 35, the stresses decay to an almost negligible value.

The layer system shown in Fig. 1 is further subdivided into a number of sublayers. The number of sublayers is concentrated in the top layers rather than those at the bottom. This decision was made in view of the fact that soil just underneath the footing is more important than layers that are rela­tively deep-seated.

At the center of each sublayer, the Boussinesq stress influence coefficient I is calculated for a square footing subjected to a uniformly distributed load. The individual sublayer influence factor Ij is obtained by dividing each of the coefficients by the sum of all the coefficients up to the depth of influence

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Excitation level, we (kg-cm)

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Static load = 1195 kg

10 20 30 40 50 60 Layer thickness , h (cm)

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3B. The equivalent stiffness of the layer system is then defined as

Keq = ZKJIJ (6)

Keq = KWj)* + K2(ZIX + K3(ZIj)hi + K4aij)hA (7)

TABLE 3. Stiffness Coefficients for Different Materials

Static load (kg) (1)

810

1,023

1,195

Excitation level (kg-cm)

(2)

1.971 3.917 7.714

1.971 3.917 7.714

1.971 3.917 7.714

Stiffness of natural soil

(kg/cm) (3)

39,201 34,454 29,357

42,595 39,137 33,048

38,601 35,054 32,923

Asymptotic Values of Stiffness

Sawdust (kg/cm)

(4)

4,820 3,869 3,525

5,357 4,800 4,500

6,250 5,357 4,900

Sand (kg/cm)

(5)

16,500 15,850 15,000

20,003 18,333 15,385

27,273 24,643 20,968

Granite chips (kg/cm)

(6)

19,231 17,500 15,000

21,296 19,231 17,692

23,076 21,429 19,231

611

Thickness ratio, h /r0

FIG. 7. Variation of Equivalent Spring Constant with Thickness of Sawdust in Two-Layered System

The values of 7, cumulatively added from zero thickness to maximum thick­ness give rise to a factor 27, that can also be used to obtain the equivalent stiffness as follows:

Keq = #,(2/,), + K2[(2lj)2 - (2/;)j] (8a)

Keq = 2K„[&j)n+l - (27,)„] (8fo)

Thus Keq is a function of footing size. Fig. 2 shows the variation of 27, with respect to h/B.

EXPERIMENTAL INVESTIGATION

The experiments conducted in this investigation were field vibration tests in the vertical mode. The area adjoining the soil mechanics laboratory of the Indian Institute of Science was chosen for this purpose. The site consisted of a thick layer of red earth that could be considered as a homogeneous semi-infinite mass. Before setting up the foundation block and vibrator, the site was cleared and the top layer removed and leveled. A reinforced concrete square footing of size 45 cm X 45 cm X 15 cm was used as the foundation

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FIG. 10. Observed versus Predicted Equivalent Spring Constant in Two-Layered Systems (Weighted Average Method)

block to which a Lazan oscillator LA-1, capable of generating eccentricities varying from 0°-140°, was rigidly attached. The oscillator was connected through a flexible shaft to a variable-speed D.C. motor that could be run at speeds up to 2,800 rpm. The speed of the motor was controlled with a var-iac. An electrodynamic sensor was used to pick up the mechanical oscilla­tions, which were measured with an analog vibration meter compatible with the pick-up. The vibration meter used could measure amplitudes of 0-1,000 fxm over five ranges; it was possible to measure up to 1 u.m in the lowest range. The frequency was recorded with a contact-type mechanical tachome­ter. The variation in static load was achieved through a number of layers of cast iron ingots of predetermined weight that were rigidly fixed to the sys­tem. The change in dynamic load was imparted by changing the eccentricity. The frequency-amplitude response curve was obtained by varying the speed of the motor and recording the corresponding frequency and displacement amplitude over a range of static and dynamic load combinations. These am­plitudes and frequencies were used in the calculations.

Tests were conducted under two series: two-layer and three-layer systems. These tests were performed in a pit of size one meter by one meter, filled

614

Spring constant, k x 10 (kg/cm) predicted

FIG. 11. Observed versus Predicted Equivalent Spring Constant in Two-Layered Systems (Odemark's Method)

and compacted with required layer materials to the required thickness to form individual layer(s) above the natural soil. The compaction was done follow­ing a uniform procedure using an earth mass compactor. The size of the pit was chosen after conducting a series of tests with a pit of smaller size (0.75 x 0.75 m) in order to ensure that the side boundary effects could be held negligible. It was ensured that reproducibility of preparation of soil systems could be achieved. The preparation of different layered soil systems was repeated each time for different layer thicknesses as desired. The thickness of the top layer in a two-layer system was increased by increasing the depth of the pit. The maximum thickness of the top layer was about 1.5 times the width of the footing. All tests were conducted for three static loads (810, 1,023, and 1,195 kg) and three eccentric moments (1.917, 3.917, and 7.714 kg-cm). A minimum of three thicknesses of each material were investigated. The actual scheme of investigation is as shown in Tables 1 and 2.

It can be seen from Tables 1 and 2 that the materials used for layering were: (1) Sawdust that had been obtained from a local sawmill; (2) river sand passing a 2.36-mm sieve and retained on a 75JJL sieve; (3) granite chips (6-mm size); and (4) natural red earth, which formed the bottommost layer.

These materials were chosen so as to ensure that they covered a wide range of material properties.

615

RESULTS

Two-Layer Systems Tests were conducted first on natural soil and then on two- and three-layer

systems at all the nine load combinations as described in Tables 1 and 2. The resonance frequency and resonance amplitude were determined from frequency-amplitude response curves. A typical response curve is shown in Fig. 3. The resonance frequency versus the thickness of the top layer (in a two-layer system) and resonance amplitude versus the thickness for sand as the top layer is shown in Figs. 4 and 5. Similar results were obtained for the other two materials, sawdust and granite chips. The experimental value of equivalent stiffness of the layered system was calculated using the fol­lowing equation:

1 Ik 2TT \j m

where / = resonance frequency; k = stiffness; and m = total mass of foun­dation and machine.

Thickness of top layer (h)

(a)

FIG. 12. Variation of Resonant Frequency with Thickness of Top Layer in Three-Layered System

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FIG. 13 . Variat ion of Equivalent Spr ing Constant with Th ickness of Sawdust a s T o p Layer in Three-Layered System

From Fig. 4, it can be seen that the resonant frequency decreases with an increase in thickness. It can also be seen that the decrease is more in the initial portion, and thereafter the variation is more gradual. The resonant frequency becomes almost asymptotic at a layer thickness of about 1.5 times the width, which can be taken as representative of a single top layer affecting the response, with the bottom layer's influence being negligible.

In order to find the individual spring constant of the top material, it was assumed that the curve relating thickness and the spring constant of the lay­ered system was a rectangular hyperbola. If this relation were to be true, a plot between h/k and h (h = thickness of layer) must give a straight line. Fig. 6 shows that this assumption is reasonably valid. The inverse of the slopes of these lines represent the value of the spring constant of the material when its thickness tends to infinity. These asymptotic values were used as the layer's spring constant if that layer were to be present as a single ho­mogeneous layer.

Figures similar to Fig. 6 have also been obtained for other materials. It was noticed that the asymptotic values of the stiffness coefficient is a func­tion of both the static load and the excitation level. This is due to the in­fluence of the confining stress and strain level. Table 3 presents the asymp­totic values obtained treating the thickness versus stiffness coefficient as a rectangular hyperbola for the three different materials for different dynamic

Excitat ion level we(kg-cm)

1-971

617

Weighted average

Static load, W = 1023 kg

04 08 V2 V6 2'0 Thickness rat io, h)/r0

2-4 2-8

FIG. 14. Variation of Equivalent Spring Constant with Thickness of Sand as Top Layer in Three-Layered System

and static load combinations. Stiffness coefficient obtained from static plate load tests yielded lesser values.

It can be seen from Fig. 5 that the amplitude also tends to remain constant beyond a thickness of about 1.5 times the width of the footing.

Prediction and Performance of Two-Layer Systems A comparison of the values of stiffness of layered systems obtained ex­

perimentally and predicted theoretically using both the weighted average method and Odemark's method has been made using graphs showing the variation of the equivalent stiffness of layered systems with the thickness of top layer. The thickness of the layer is represented as a nondimensional pa­rameter in terms of the radius of a circular base with an equivalent area. This was required because the Odemark's method demands a circular base. Figs. 7-9 show typical plots for various materials and static loads. Fig. 7 is for sawdust as the top layer, whereas Fig. 8 and 9 are for sand and granite chips as the top layer, respectively. It can be seen that both experimentally and theoretically, the stiffness reduces with an increase in layer thickness. This is because the natural soil has a stiffness higher than that of any of the individual top layer materials (Table 3). It can also be seen that the decrease is greater in sawdust than in sand and granite chips. This is to be expected

618

Top layer Middle layer o Saw dust Sand

0-04 0-08 0-12 0-16 0-20 0-24 0-28 Spring constant, kx iff5(kg/cm) predicted

FIG. 15. Observed versus Predicted Equivalent Spring Constant in Three-Lay­ered Systems (Weighted Average Method)

since! the introduction of a very weak layer at the top is bound to reduce the equivalent stiffness more. It can also be seen that the weighted average method predicts more accurately than Odemark's method in all the three cases. Figs. 10 and 11 show a comparison of the accuracy of prediction by the two meth­ods for the entire range of results obtained. It can also be seen that in the, case of sand, the weighted average method predicts values that are very close to the observed ones, whereas there is some error in the case of sawdust. In comparison, Odemark's method overpredicts the values for all three ma­terials, the worst case being for sawdust.

Three-Layer Systems The behavior of three-layer systems under vertical dynamic loading was

also studied with sawdust and sand forming the top two layers overlying the natural soil. Experiments were also conducted with sawdust and sand inter­changed to examine the effect of order of layering. Table 2 shows the scheme of investigation carried out.

Figs. 12(a and b) show typical results regarding the variation of resonant frequency against thickness of the top layer. The extreme points refer to two-layer systems. It shall be mentioned that the total thickness of the top two

619

Top layer Middle layer

o Saw dust Sand

"~&C& O08 QA2 006 020 074 0-28

Spring constant, k x 10"5(kg/cm) predicted

FIG. 16. Observed versus Predicted Equivalent Spring Constant in Three-Lay­ered Systems (Odemark's Method)

layers was kept constant at 69 cm. Thus an increase in thickness of the top layer requires a decrease in thickness of the intermediary layer. In Fig. 12(a), the decrease of resonant frequency is due to an increase in thickness of the sawdust and a decrease in thickness of the intermediary sand layer. In con­trast, in Fig. 12(b), it is seen that as the thickness of the top layer increases (thickness of sand increasing and thickness of sawdust decreasing), the res­onant frequency increases. This is expected as the sand layer has a higher stiffness than sawdust (Table 3), resulting in a higher resonant frequency as the thickness of sand increases.

Prediction and Performance of Three-Layer System A comparison has been made here between the results of the two methods,

i.e., the weighted average method and Odemark's method, and the experi­mental results of three-layer system. Fig. 13 shows typical results of the variation of stiffness with respect to thickness for sawdust over sand over natural soil for a static load of 1,023 kg. Fig. 14 shows the same for sand over sawdust over natural soil. Similar results have been obtained for other

620

A-R.C. slab o - Sand passing (No. 7 sieve) x - Granite chips (6 mm)

Static load , w(kg)

FIG. 17. Resonant Frequency versus Static Load for Equal Thickness (7.5 cm) of Different Materials as Top Layer in Two-Layered System

static loads. In Fig. 13, it can be seen that as the thickness of sawdust in­creases, the spring constant decreases. This is because sawdust is a weaker material than sand. The experimental results tend to be asymptotic with an increasing thickness of sawdust. Theoretical prediction by both the weighted average method and Odemark's method has been shown in Fig. 14. The weighted average method can be seen to predict better than Odemark's in this case as well. However, both methods predict higher values of equivalent stiffness than was obtained experimentally.

Fig. 14 shows typical results for the behavior of the layered system wherein the top layer is sand and the intermediary layer is sawdust. It can be seen that the equivalent stiffness increases with an increase in the thickness of the sand layer at top. Another observation is that the experimental results do not show any asymptotic trend even at the maximum thickness tested. Further, the experimental points show that the influence of dynamic load on equivalent stiffness increases as the thickness of sand increases. In this case, Odemark's method predicts better than the weighted average method. Sim­ilar results have been obtained for other static loads. Figs. 15 and 16 show predicted versus observed values for the three-layer system. The weighted average method predicts marginally better when the top layer is sawdust, whereas Odemark's method predicts better when the top layer is sand. Both methods, however, show a higher degree of scatter in the case of sand over sawdust.

621

EFFECT OF DIFFERENT MATERIALS BETWEEN FOUNDATION AND NATURAL SOIL

In order to study the influence of a variety of materials of different stiff­nesses, a series of experiments were carried out keeping a constant thickness of 7.5 cm as an intermediary layer between the foundation block and the natural soil. The experiments were carried out similarly to the preceding description with a foundation block of 45 cm X 45 cm x 15 cm. Experi­ments were carried out with three different static loads and excitation levels. The various materials used for the intermediary layer were sawdust, vibro-cork, 6-mm granite chips, sand passing a no. 7 sieve, a reinforced concrete slab, and black cotton soil.

Fig. 17 shows the variation of resonant frequency with static loads for different excitation levels. The variation of resonant frequency with static load is also shown for a foundation resting on natural soil without any in­termediary layer. It can be seen that a small thickness of 7.5 cm (1/6 B)

130

110

90

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we = 7-7U kg-cm

1600 800 1000 1200 Static load , w (kg)

x - Sawdust o - Vibro cork (50mm) * - Granite chips (6mm) D- Sand passing (No. 7 sieve). A - R.C.slab 9 - B.C. soit (w=36-4)*b=1'732

Natural soil g/cc

1200 Static load , w (kg)

FIG. 18. Maximum Displacement Amplitude versus Static Load for Equal Thick­ness (7.5 cm) of Different Materials as Top Layer in Two-Layered System

622

m the intermediary layer can change the resonant frequency by almost two­fold. Both the static load and dynamic load have a marginal effect.

Fig. 18 shows the behavior of resonant amplitude with variation in static load using different materials as the intermediary layer. The effect of an intermediary layer of small thickness has a very significant influence on the resonant amplitude. The variation of resonant amplitude is of the order of 600%. These results demonstrate the possibility of changing the resonant amplitude toward the desired level by introducing a suitable intermediary layer.

CONCLUSIONS

A new but approximate method to obtain a realistic estimate of the equiv­alent stiffness of a layered system, when its individual properties are known, has been proposed. This method, the weighted average method, stresses the location and extent of the component layers relative to the foundation base. A series of simple experiments were performed to study the theoretical pre­dictions. The results obtained using different materials of different thick­nesses were predicted analytically using both Odemark's and the newly pro­posed theoretical procedures. A comparison was made of the accuracy of prediction between the two methods. The inherent advantages in the newly proposed method are demonstrated by its applicability to both two- and three-layer systems.

The importance of layering and its effect on the various parameters, e.g., frequency and amplitude, have been shown through a series of graphs. The effect of the presence of different materials over natural soil has also been developed. This investigation shows that by using a stronger material of small thickness just beneath the foundation, one can vastly improve the per­formance of the foundation and thereby obtain an economical design.

APPENDIX I. REFERENCES

Akai, et al. (1971). "Model studies on stress distribution in layered systems." Proc. JSCE, 185, 83-93.

Arnold, R. N., Bycroft, G. N., and Warburton, G. B. (1955). "Forced vibrations of a body on an infinite elastic solid." Trans. ASME, 77(3), 391-401.

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APPENDIX II. NOTATION

The following symbols are used in this paper:

B = width of footing; Ee = equivalent modulus of elasticity of layered system;

E 1,2,3 ~ modulus of elasticities of layers 1, 2, and 3, respec­tively;

Fco = nondimensional displacement factor; / = frequency;

fmr = resonance frequency; h = thickness of layer;

^i,2,3 = thickness of layers 1, 2, and 3, respectively; / = Boussinesq's stress factor; /,• = individual sublayer factor; k = spring constant or stiffness;

ke,keq = equivalent spring constant of layer system; k„ = spring constant of nth layer;

i,2,3 = spring constant of layers 1, 2, and 3, respectively; N2,N2,N3,N3 = terms in Odemark's expressions;

r0 — radius of footing; and Xlj = cumulative influence factor for layer.

APPENDIX III. UNIT CONVERSION FACTORS

To convert

kg kg-cm kg/cm g/cc cm RPM

to

N N-mm N/mm N/mm3

mm Hz

Multiply by

9.81 98.1 0.981 9.81

10 1/60

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