FOOTING RESPONSE TO HORIZONTAL VIBRATION

By M. V. Nagendra1 and A. Sridharan2

INTRODUCTION

Resonant frequency and resonant amplitude are the two important cri­teria involved in the design of foundations subjected to periodic loading. The effect of the type of contact pressure distribution and the displace­ment condition are of importance while predicting the dynamic response of footings (2,4,7,9). Elastic half-space model has been used by many investigators (1,3,5,6,11) to predict the dynamic response. Bycroft (1) ob­tained solution for the response of a rigid circular footing subjected to t frequency dependent horizontal excitation, considering weighted aver­age displacement, assuming that the vertical displacement due to hori­zontal load is zero. Luco and Westmann (6) obtained the Fredholm in­tegral equations governing the response of a rigid circular footing, assuming that the vertical stress due to the horizontal load is zero. It can be seen from literature that for horizontal vibration, solutions are i available only for rigid base shear distribution for weighted average dis- \ placement condition. However, because the relative rigidity of footing ! and soil is bound to vary, different types of shear distributions are in­evitable in actual practice, as have been well established for vertical vi- ] bration. Therefore, in this paper, solutions have been obtained for the cases where solutions are not available, considering three types of con­tact shear distributions; rigid base, uniform and parabolic and three dis­placement conditions; central, average and weighted average. It has been j assumed that the vertical stress developed due to horizontal dynamic force is zero (6,11).

Analysis.—From the equations of motion of a linear elastic medium, in cylindrical coordinates, the solution for the displacement compo­nents, u, v, and w in r, 0 and z directions was obtained by Sezawa (10). \ Considering the case where an oscillator is acting upon the surface of a semi-infinite solid, and if the oscillator is replaced by a contact shear force of amplitude, Px, with frequency, co, acting at the surface (z = 0), the boundary conditions for the three types of contact shear distribution I may be written as follows: I

(a) Rigid base contact shear distribution:

az = 0 (la)

Pxeiat !

T* = o—71 ~s c o s 8 for /• < a ^

'Research Scholar, Dept. of Civ. Engrg., Indian Inst, of Science, Bangalore 560 012, India.

2Assoc. Prof., Dept. of Civ. Engrg., Indian Inst, of Science, Bangalore 560 012, India.

Note.—Discussion open until September 1, 1984. To extend the closing date one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for i review and possible publication on January 28, 1982. This paper is part of the Journal of Engineering Mechanics, Vol. 110, No. 4, April, 1984. ©ASCE, ISSN \ 0733-9399/84/0004-0648/$01.00. Paper No. 18704. \

648

J. Eng. Mech. 1984.110:648-654.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Syra

cuse

Uni

vers

ity L

ibra

ry o

n 04

/13/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

= 0 for r > a (lb)

Pxeiu"

T7o = : 7- sin 0 for r < a z9 2-na(a2-r2)

= 0 for r > a (lc) (b) Uniform Contact Shear Distribution:

ffz =• 0 (2a)

Tzr = , cos 0 for r < a

= 0 for r > a (2b)

Pxe'-" TZ8

= r s i n i for r<a TTfl

= 0 for r > a (2c)

(c) Parabolic Contact Shear Distribution:

<rz = 0 (3a)

2Px(a2 - r2)eiwt

T2r = 2 c o s 9 f° r r < a

ir a = 0 for r > a (3b)

2Px(a2 - r V "

TZ6 = T sin0 for r<a TTfl

= 0 for r > a (3c)

in which a = the radius of area over which the load is assumed to be acting. Expressing the three types of contact shear distributions as a Fourier-Bessel integral (1) and substituting the boundary condition that the vertical stress CT2 due to horizontal load = zero (6,11), the displace­ment, U at any point (r, 0) on the surface (z = 0), in the direction of horizontal force Px is

U = u cos 0 + v sin 0 (4a)

f Px M(xa)\£K2 a lh(xr) in which u= Ji{xr)

J0 TxGa x \_F(x) Br p r

• e'°" cosQdx (4b)

f Px M(xa)\£k2h(xr) 1 d 1 M .

J0 uGa x \P{x) r p dr J

in which M(xa) = sin (xa)/2 for rigid base, = h(xa) for uniform and = 4]2(xa)/xa for parabolic contact shear distributions, respectively. F(x) = (2x2 - k2)2 - 4apx2, p = (x2 - h2)1'2, a = (x2 - k2)1'2, k = pw2/G, h2

= pco2/(^ + 2G), X = Lame's constant, G = shear modulus, and p = mass density of the semi-infinite solid.

649

J. Eng. Mech. 1984.110:648-654.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Syra

cuse

Uni

vers

ity L

ibra

ry o

n 04

/13/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Displacement Conditions.—Using Eq. 4 the central displacement, Uc, is determined by putting r = 0, 6 = 0. The average displacement, Ua and the weighted average displacement, Uw are evaluated by the ex­pressions

4 f r'2

!i„ = — U-rdr-dQ (5a) ir« Jo Jo

fa rir/2

and UW = A \ | p^U-rdr-dQ. (5b)

in which px = 1/2 in? (a2 - r2)1 / 2 for rigid base, = 1/TTA2 for uniform, and = 2(fl2 - r2)/irfl4 for parabolic contact shear distributions, respectively. By changing the order of integration, simplifying and substituting x = kn, a0 = ka, and h/k = s and simplifying, the displacements Uc, Ua and Uw are given by

Uc-Pxe'°" 2 _ -IN 1/2 (n2 - 1)

2-nGaJo I F(n) (n2 - 1) 1/2

u. = -P r e"

LT„ = :

luGa j 0

Pre"°'

1/2 (*' - 1)

F(n) (n2 - 1)1/2.

V - 1)1/2 1 2<i7GflJ0 [ F(n) ( n 2 - l ) 1 / 2 _

M(a0n)dn (6a)

2-M(a0n)-J1(a0n)

a0n

2-[M(a0n)]2

a0n

•dn

dn

(6b)

(6c)

in which F(n) = (2n2 - l ) 2 - 4n2(n2 - s2)1/2(n2 - 1)1/2, and nondi-

Reissner's method (8 ) Contour integration

Frequency f a c t o r , a 0

FIG. 1.—Comparison of fx and f2 Values Determined by Reissner's Method (10) and Contour Integration

650

J. Eng. Mech. 1984.110:648-654.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Syra

cuse

Uni

vers

ity L

ibra

ry o

n 04

/13/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

TABLE 1.—Values of bv lor Different Poisson's Ratios

M-(1)

0 1/4 1/3 1/2

s2

(2)

1/2 1/3 1/4

0

b0

(3)

- 2 - 7 / 4 - 5 / 3 - 3 / 2

bi

(4)

- 9 / 8 - 1 3 / 1 6 - 5 3 / 7 2 - 5 / 8

b2

(5)

- 6 7 / 6 4 -125/192 -1975/3456 - 1 5 / 3 2

mensional frequency factor, a0 = wa(p/G)1/2. A comparison (Fig. 1) of the displacement functions, /i and f2 evaluated for rigid base pressure distribution, weighted average displacement, by using Reissner's (8) method and by contour integration (1), for vertical vibration showed very good agreement. For reasons of simplicity, Reissner's (12) method has been adopted in this study, to evaluate the displacement functions, /] and f2. By splitting the integral and separating the real and imaginary parts, Eq. 6 can be expressed in the form

u = fa(h + ih) (7)

The displacement function, / i , can be expressed as a series function of nondimensional frequency factor, a0 in the form (12)

h 2 T T ^ 0

K-Gx 7

(8)

in which bv and Gv are constants. The values of bv and Gv for, = 0, 1 and 2 are evaluated and presented in Tables 1 and 2. The value of bv is

TABLE 2.—Value of G sure Distributions

Displacement condition

(1)

Central"

Average

Weighted average

„ for Different Displacement Conditions and Contact Pres-

Contact pressure distribution

(2)

Rigid base Uniform Parabolic Rigid base

Uniform

Parabolic

Rigid base

Uniform

Parabolic

Go (3)

TT/4

1 4/3 TT/4

8

3TT

128

45TT

IT/4 8

3TT

1,024

315ir

G1

(4)

- T T / 8

- 1 / 3 - 4 / 1 5

- 5 T T / 3 2

- 6 4 •

45TT

-2 ,048

1,575TT

- I T / 6

- 6 4

45TT

-16,384

14,175TT

G2

(5)

ir/96 1/45

4/315 7ir/256

1,024

4,725TT

16,384

99,225-rr TT/30

1,024

4,725TT

131,072

1,091,475TT

"Values of Gv for central displacement are taken from Sung (1953).

651

J. Eng. Mech. 1984.110:648-654.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Syra

cuse

Uni

vers

ity L

ibra

ry o

n 04

/13/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

TABLE 3.—Values of the Coefficients Xt, X2, X3

Conditions and Contact Shear Distributions

Displacement condition

d) Central

Average

Weighted average

Contact pressure distribution

(2)

Rigid base Uniform Parabolic

Rigid base Uniform Parabolic

Rigid base Uniform Parabolic

X1

(3)

1 1 1

1 1 1

1 1 1

for Different

X2

(4) -1 /6 -1 /8 -1/12

-7/24 -1 /4 -5/24

-1 /3 . - 1 /4 -1 /6

Displacement

x3 (5)

1/120 1/192 1/384

11/320 5/192 7/384

2/45 5/192 7/576

a function of Poisson's ratio only and the value of G„ is a function of type of contact shear distribution and displacement condition. On nu­merical integration, the displacement function, f2, can be expressed as

f2 = 0.113688(Xi^o - 0.064197(X2)ag + 0.046855(X3)a50 (9a)

f2 = 0.106076(X^flo - 0.059227(X2)a^ + 0.044019(X3)a^ (9b)

f2 = 0.102290(X1)tf0 - 0.057487(X2)«g + 0.043267(X3)a50 (9c)

f2 = 0.094542(X!)fl0 - 0.054933(X2)a3

0 + 0.042339(X3)a5

0 (9d)

•o c a

M = O . O u=0.25 11=0.33 11=0. 5

\ \ ^ - / l = 0

\ ^ / 1 = 0.

^ ' 2

) 25 33 5

0.5 1-0 1-5

Non d imens iona l frequency f a c t o r , d 0

FIG. 2.—Relationship Between Displacement Function fx, f2 and Frequency Fac­tor, «„

652

J. Eng. Mech. 1984.110:648-654.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Syra

cuse

Uni

vers

ity L

ibra

ry o

n 04

/13/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

TABLE 4.—Values of Stiffness Coefficients for Horizontal Mode

Displacement condition

0)

Central

Average

Weighted average

Contact Shear Distribution

Rigid base (2)

8G«

2 - j i (Mindlin, 8)

8Gfl

2-M-(Mindlin, 8)

8Gfl 2 - |x

(Mindlin, 8)

Uniform (3)

2irG«

2 — |x (Authors)

3n2Gfl

4(2 - V ) (Authors)

3TT2G«

4(2 - n) (Authors)

Parabolic (4)

3TTG«

2(2 - p.) (Authors) 45TrzGfl

64(2 - v) (Authors) 315Tr2Ga

512(2 - M-) (Authors)

for four values of Poisson's ratio, u., viz., 0.00, 0.25, 0.33 and 0.50, re­spectively. The values of the constants Xi , X2 , X3 which are functions of the type of shear distribution and displacement condition are pre­sented in Table 3. Fig. 2 presents the typical results of variation of dis­placement function ft, /2 with nondimensional frequency factor, a0 for weighted average displacement condition—uniform shear distribution. Similar results could be obtained for other cases of displacement and shear distribution conditions using Tables 1, 2 and 3 and Eqs. 8 and 9.

The value oifi for aQ = 0, represents the factor -Ga/K, where K is the static stiffness coefficient. Table 4 presents the values of K for three types of contact shear distribution and displacement conditions.

CONCLUSIONS

For the prediction of response of footings subjected to horizontal vi­bration, different types of contact shear distributions and displacement conditions are to be considered. Solutions using elastic half-space theory are not available for all the cases of shear distribution and displacement condition. In this paper, solutions have been obtained for the cases in which solutions are not available and the relevant coefficients have been presented in Tables which could be used in the appropriate equations for the prediction of dynamic response. Spring constants have been eval­uated and tabulated for different displacement and shear distribution conditions.

APPENDIX.—REFERENCES

Bycroft, G. N., "Forced Vibration of a Rigid Circular Plate on a Semi-infinite Elastic Space or on an Elastic Stratum," Philosophical Transactions of the Royal Society, London, Series A, Vol. 248, No. 948, 1956, pp. 327-368. Chae, Y. S., Hall, J. R., Jr., and Richart, F. E., Jr., "Dynamic Pressure Dis­tribution Beneath a Ring Footing," Proceedings, Sixth International Confer­ence on Soil Mechanics and Foundation Engineering, Montreal, Canada, Vol. 2, 3/5, Sept., 1965, pp. 22-26. Gladwell, G. M., "Forced Tangential and Rotatory Vibration of a Rigid Cir-

653

J. Eng. Mech. 1984.110:648-654.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Syra

cuse

Uni

vers

ity L

ibra

ry o

n 04

/13/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

cular Disc on a Semi-infinite Solid/' International Journal of Engineering Science, Pergamon Press, Vol. 6, 1968, pp. 591-607.

4. Housner, G. W., and Castellani, A., "Discussion of Comparison of Footing Vibration with Theory," Journal of Soil Mechanics and Foundations Division, ASCE, Vol. 95, No. SMI, Proc. Paper 6324, Jan., 1969, pp. 360-364.

5. Hsieh, T. K., "Foundation Vibrations," Proceedings, Institution of Civil En­gineers, London, Vol. 22, Paper No. 6571, June, 1962, pp. 211-226.

6. Luco, J. E., and Westmann, R. A., "Dynamic Response of Circular Foot­ings," Journal of Engineering Mechanics Division, ASCE, Vol. 97, No. EMS, Proc. Paper 8416, Oct., 1971, pp. 1381-1395.

7. Moore, P. J., "Calculated and Observed Vibration Amplitudes," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SMI, Proc. Paper 7789, Jan., 1971, pp. 141-148.

8. Reissner, E., "Stationare, Axial Symmetriche durch eine Schittelnde masse erregte Schwingungen einess homogenen elastichen halbraumes," Ingen-ieur—Archiv., Vol. 7, Part 6, Dec, 1936, pp. 381-396.

9. Richart, F. E., Jr., and Whitman, R. V., "Comparison of Footing Vibration with Theory," Journal of Soil Mechanics and Foundations Division, ASCE, Vol. 93, No. SM6, Proc. Paper 5568, Nov., 1967, pp. 143-168.

10. Sezawa, K., "Further Studies on Rayleigh Waves Having Some Azimuthal Distribution," Bulletin, Earthquake Research Institute, Tokyo, Japan, Vol. 6, No. 2, 1929, pp. 1-18.

11. Veletsos, A. S., and Wei, Y. I., "Lateral and Rocking Vibration of Footings," Journal of Soil Mechanics and Foundation Division, ASCE, Vol. 97, No. SM9, Proc. Paper 8388, Sept., 1971, pp. 1227-1248.

MINIMUM SCF FOR TWO NEIGHBORING HOLES IN DISKS BY PHOTOELASTICITY

By N. K. Naik1 and K. Rajaiah2

INTRODUCTION

In engineering structures, openings of circular or other forms gener­ally occur in groups. Their presence alters the stress distributions in their neighborhood invariably increasing the stress concentrations consider­ably. Examples of this kind are found in various engineering construc­tions and one of the earliest photoelastic investigations on this problem in ship structures is given in Ref. 5. Minimizing the stress concentrations in these structures is an important engineering problem.

'Lect., Aeronautical Engrg. Dept., I.I.T., Bombay-400 076, India. 2Prof., Aeronautical Engrg. Dept., I.I.T., Bombay-400 076, India. Note.—Discussion open until September 1, 1984. To extend the closing date

one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for review and possible publication on June 3, 1983. This paper is part of the Journal of Engineering Mechanics, Vol. 110, No. 4, April, 1984. ©ASCE, ISSN 0733-9399/ 84/0004-0654/$01.00. Paper No. 18704.

654

J. Eng. Mech. 1984.110:648-654.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Syra

cuse

Uni

vers

ity L

ibra

ry o

n 04

/13/

13. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.