Milos Novak

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 3, 79-96 (1974)

EFFECT OF SOIL ON STRUCTURAL RESPONSE TO WIND AND EARTHQUAKE*

MILOS NOVAKt

Faculty of Engineering Science, The University of Western Ontario, London, Canada

SUMMARY An approximate analytical approach is presented which makes it possible to consider soil properties and footing embedment in the analysis of the response of structures to external excitation such as wind and earthquake. The approach is based on modal analysis and the definition of' stiffness and damping due to soil pertinent to each vibration mode. The approach also facilitates the analysis of coupled motions of a footing alone. The analysis of a tall chimney for the effects of gusting wind, vortex shedding and earthquake is used as an example.

INTRODUCTION

The effect of soil on the response of structures depends on the properties of soil, properties of the structure and the nature of the excitation. These relations have been investigated by a number of researchers, usually under the heading of soil-structure interaction. In any case, the response can be solved directly, e.g. in terms of Fourier analysis or other methods.24 The direct approach has been used almost generally.

An alternative approach can be based on modal analysis. The latter approach, which is quite commonly used in cases of structures on rigid foundations, has been largely ignored in the soil-structure interaction problems. It is only recently that the advantages of modal analysis have been put forward even for this category of problems by Jennings and Bielak?, Roesset, Whitman and Dobry,' Chopra and Gutierrez? Vaish and Chopra?, Itohlo and the auth0r.l

Modal analysis is familiar to practising engineers, gives a clear idea about the nature of the response, can be considerably simpler than the direct solution and in some cases represents the only adequate method because the damping originating from other sources can be most conveniently described as a function of individual vibration modes. The aerodynamic damping of tall buildings and suspension bridges is an example of the latter case. Finally, the structural response is quite often dominated by one modal component with the contribution of other modes being negligible, and then the modal analysis offers particularly spectacular simplicity.

Studies concerning rigid foundations have shown that embedment beneath the surface of the ground is one of the factors most affecting the stiffness and damping derived from soil. Indeed, the theory and field experiments indicate that with most vibration modes, no realistic prediction of soil reactions can be made if embedment is neglected and the footing considered as a surface one.l1-l3 Therefore, embedment is considered in the approach outlined in this paper.

The soil reactions to be included in the analysis of structures can be obtained from solutions concerning embedded footings.

The embedded footings represent a very difficult problem to solve in a rigorous analytical way; only some special cases seem to be analytically accessible (e.g. Tajimi14). In general, the finite element technique15 or other discretizing techniqueP appear very useful in solving the problem. However, it would be a loss to entirely discard the already numerous and sophisticated solutions of the surface foot ing~l ' -~~ and others

~~

* Based on the paper presented to the ASCE National Structural Engineering Meeting, San Francisco, 1973.' t Professor of Engineering Science.

Received 19 November 1973 Revised 25 February 1974

@ 1974 by John Wiley & Sons, Ltd. 79

80 MILOS NOVAK

because of the possible effect of embedment. Instead, it may often be quite sufficient, with all the other uncertainties, to apply an approximate correction for the effect of embedment to the solutions of surface footings. An approximate analytical approach of this kind is applied in this paper. The approach is stream- lined and brought to a form suitable for modal analysis of structures and footings.

SOIL REACTIONS

The approximate analytical description of soil reactions is based on the assumption that the soil reactions acting at the footing base are equal to those of a surface footing, and the reactions acting on footing sides are equal to those of an independent layer overlying the soil beneath the footing base. As well, this overlying layer is considered to be composed of a series of infinitesimally thin independent layers. The latter assumption makes it possible to readily solve more complicated motions involving non-uniform displacements of footing sides.

Such an approach to footings was first adopted by B a r a n ~ v ~ ~ and has been extended by the writer and his associates, Beredugo and Sachs, to include torsion, layering and coupled vibration modes, and also to develop formulas for stiffness and damping constants of soil directly applicable to embedded foot ing~. l l -~~ The formulas can be introduced into the analysis of structures as they are compatible with standard approaches of structural dynamics. Thus, a stiffness (spring) coefficient kij represents a force applied at the reference point (e.g. centre of gravity) in direction i to produce a unit displacement in directionj. Similarly, a damping coefficient cij will define a damping force produced in direction i due to a unit velocity in directionj. The location of the reference point is arbitrary. (Mass moment of inertia and displacements relate to the same point.)

For cylindrical embedded footings, the equivalent stiffness and damping constants are summarized below, in a somewhat modified form, for use in this paper:

For vertical vibration w, the stiffness constant

(1)

and the damping constant

For torsional vibration 5 the stiffness constant

and the damping constant

For coupled horizontal translation u and rocking i+h (Figure l), the stiffness constants are

EFFECT OF SOIL ON STRUCTURAL RESPONSE TO WIND AND EARTHQUAKE 81

and the damping constants are

Figure 1 . Translation and rocking components of coupled motion and related stiffness and damping coefficients of embedded footings

Analogous formulas for coupled motions involving trans!ation, rocking and torsion are given in Reference 13.

In the above equations, G and p are the shear modulus and density of soil beneath the footing base, G,, ps are the shear modulus and density of soil adjacent to footing sides (backfill), r, = radius of a cylindrical footing or equivalent radius of a rectangular footing,$ 8 = Z/r, = relative embedment depth, 1 = embedment depth, z, = height of reference point above the footing base.

Parameters C relate to footing base reactions and typically are

where fl and fi are displacement functions of Reissner’s type of argument a, = r, w ,/(p/G). Parameters S relate to side reactions and their typical form is

and

1 4 3 --- c2 - a, JE+ Y ; (9)

Here, Jo(a,), Jl(a,) are Bessel functions of the first kind of order zero and one respectively, and Yofa,), ?(ao) are Bessel functions of the second kind of order zero and one. For side reactions, a, = r, w ,/(p,/G,). Para- meters with subscript 1 relate to stiffness and parameters with subscript 2 relate to damping.

Parameters C and S are shown in Figures 2 and 3 for several values of Poisson’s ratio given in brackets.

$ Some guidelines for the choice of the equivalent radii of rectangular footings can be found in References 18 and 24.

82 MILOS NOVAK

r

I I I I I l l I I I I I I 0.5 1.0 1.5

DIMENSIONLESS FREQUENCY a,

Figure 2. Examples of stiffness parameters C, and S,. (Poisson's ratio given in brackets)

12

10

Iv)" ;u8

10 v) LL w t- Y 6

2 I 4

a

a Q

(3

Q z 0

2

0 0.5 1.0

DIMENSIONLESS FREQUENCY 0,

1.5

Figure 3. Examples of damping parameters Ca and Sa. (Poisson's ratio given in brackets)


Parameters C shown were calculated with Bycroft's functions &.17 For other values of Poisson's ratio and a larger frequency range, parameters C, and C, can be obtained from Veletsos and Wei21 with whose notation

8 8 2-v 2-v Cul=-kl, CUa=-c1

and

Other parameters S are given in References 11-13 where the damping parameters are given in the form of S2 = a, S2 and C2 = a, C2.

Parameters C and S are frequency dependent and so are consequently stiffness constants k and damping constants c. However, it can be seen from Figures 2 and 3 that most parameters C,, S, and C2 and fZ do not vary with frequency too much, and for practical purposes can often be considered approximately constant at least over a certain frequency range of interest. (It may be noted that these parameters vary with frequency much less than displacement functionsf,,,.)

Several values of such approximate constant parameters are given in Table I and listed for cohesive and granular soils.

Table I. Stiffness and damping parameters

Motion Soil Side layer Half-space ~~

Sliding Cohesive S,, = 4.1 5,s = 10.6 C,, = 5-1 cus = 3.2 Granular s,, = 4 0 s,, = 9-1 C,, = 4.7 cu* = 2.8

C,, = 4.3 c,, = 0.7 Sq,, = 2.5 3, = 1.8 Rocking Cohesive Granular c,, = 3.3 c,, = 0.5

s,, = 10.2 s,, = 5.4 Torsion Cohesive Granular

- Cm = 7.5 C a = 6.8 S,, = 2.7 3, = 6.7 Vertical Cohesive

Granular C, = 5.2 cws = 5.0

The above definition of stiffness and damping constants is very versatile. The effect of layering can be included through the substitution of strata reactions into expressions for C. The effect of backfill can be accounted for by using ps< p and G,< G. If the bond between the footing sides and backfill is not reliable, intuitively reduced values of parameters S may be used.

With the above equivalent stiffness and damping constants, the response of footings in the vertical, torsional and coupled vibration modes can be directly calculated without difficulty. Either frequency dependent or independent parameters can be used. With proper choice of the constant parameters, response curves can easily be obtained that only slightly differ from those computed with variable parameters.

With piles, formulas analogous to equations (1) to (6) are available in Reference 25.

MODAL ANALYSIS IN SOIL-STRUCTURE INTERACTION PROBLEMS

The stiffness and damping properties of soil can be introduced into the analysis of structures and footings simultaneously to directly yield the response. This has been done by Parmelee, Perelman and Lee; ParmeleeYzs Rainer:' Whitman, Protonotarios and Nelsonm and others, and analysed in detail by Jennings and Bielak.6

With multi-degree of freedom systems, such as tall buildings or chimneys, it appears more convenient to consider the effect of soil stiffness separately from the damping, which can be done in terms of modal analysis (superposition).

84 MILOS NOVAK

The rigorous application of modal analysis to soil-structure interaction problems features considerable difficulties because of the theoretical lack of classical normal modes due to the frequency variable parameters of soil as recently discussed by Jennings and Bielak.s However, there are several circumstances that seem to justify some approximations in the development of modal analysis for use in soil-structure interaction. Field experiments indicate that orthogonal vibration modes do exist with real structures, in many cases the response is dominated by one mode and finally, the elastic half-space is a very approximate model of the real soil anyway.

The equations of vibrations of a linear structure can be written in matrix form

in which [m],,[c] and [k] are square matrices of mass, damping and stiffness; and {u} and {P} are vectors of displacement and excitation. Only velocity proportional damping is considered here because it reasonably represents the radiation (geometric) damping in the elastic half-space. Internal (hysteretic) damping of soil is neglected which appears advisable with embedment included.

If damping and excitation is omitted in the first step of the analysis and the soil stiffness considered frequency independent, equation (12) determines the orthogonal modes of free undamped vibrations and the corresponding frequencies. The modes can be written as [a] with the individual modes listed as columns and used to describe the response in terms of generalized co-ordinates, q,

{u> = PI{q> (13)

(14)

Substitution of equation (1 3) in (12) yields the equations for generalized co-ordinates

FrI,Cii>+ [*lTICl [*I {a>+ [IcIa{cr) = [*IT{P> in which [MI, and

Now, the difficulty is encountered that with damping of soil included, the damping matrix [ ~ ] T [ c ] [a] always has off-diagonal terms. Consequently, equation (14) does not split into independent equations for generalized co-ordinates and the modal analysis loses one considerable advantage. To overcome this difficulty various not quite simple procedures can be used to diagonalize the damping matrix (29), or some authors prefer to work with damped vibration modes from the beginning.6* lo To get around these difficulties, the following energy consideration appears suitable and is used in this paper.

Damping of the structure due to soil Assume that the undamped vibration modes and frequencies of a structure (Figure 4) have been determined

with the soil flexibility taken into account as described by the stiffness constants. The inclusion of soil

are diagonal matrices of generalized mass and generalized stiffness.

... ..

.,. &--

U I I

Figure 4. Structure vibrating in j th natural mode

t 1 ]d represents a diagonal matrix.


flexibility in the analysis cf free undamped vibration does not represent any particular problem, as most computer programs allow for it or can be adapted to that end (e.g. Reference 30).

Assume that each section of the structure is free to undergo horizontal translation ui, rotation in vertical plane y5i and torsion around vertical axis Ci, and that the structure and its footing harmoniously vibrate in the j th natural mode with natural frequency wj. The natural modes and frequencies can be approximately considered equal to those of an undamped system.

During such vibrations in the j th mode the footing undergoes translation ulj, rocking #lj and torsion Clj. The work done during a period of vibration T = 277/wi by the damping forces P ( j ) is, in general,

This formula can be applied to the work done by the footing's equivalent damping (dashpots) (Figure 1) during the vibration of a structure in a natural mode.

The damping forces P and moments M acting at the footing's reference point during harmonic motions

ulj(t) = ulj sin wj t, +lj(t) = t,hlj sin wj t, Clj(t) = sin wj t are

i P(li) = c,, ii = c,, Ulj wj cos wj t

M ( 4 ) = C$$ 4 = C$$ wj cos w j t P ( 4 ) = C,$ 4 = Cr$ +lj wj cos wj t

M(U) = c*, li = C,$ U1j wj cos w j t

M ( [ ) = C" 5 = cgc CljWjC0SWj t

(In this approach cPr = c,$, which need not be generally quite true.21*15) The total work done by these forces during a period T is, according to equation (15),

W = IOT c,, u s w: cos2 wj t dt + +z w: cos2 oj t dt + [ex$ t,hlj ulj w: cos2 wj t dt

The maximum potential energy of the whole structure can be calculated as maximum kinetic energy and is

in which mi = mass, Ii = mass moment of inertia in the vertical plane, Ji = mass moment of inertia about vertical axis and n = number of masses.

Then the damping ratio of the structure due to the geometric damping of the soil is defined for the response i n the j t h mode as Di = W/(477L) which is

in which generalized mass

Here, c = soil equivalent damping constants given by equations (4) and (6), uij, (Gij and Cij = modal displacements taken in arbitrary scale, ulj, #lj and Clj = modal displacements of the footing and wj = the j t h natural frequency of the structure on flexible foundation.

86 MlLOS NOVAK

The damping due to soil can be added to the other components of damping to yield the total damping D$ and the response ui, y$ and ci to external excitation

I where r = number of natural modes (degrees of freedom). The generalized co-ordinates are obtained from independent equations

in which the generalized force producing response in the j th mode only is

where Q, Mi and Ti are the forces and moments of excitation acting in the corresponding direction. (Mi = moment acting on mass i while Mi is the generalized mass of mode j . )

With vertical vibrations of a structure, the same procedure can be applied to obtain the modal damping due to soil. Assume a purely vertical vibration mode characterized by modal displacements wij of individual masses in vertical direction and by natural frequency oj. In this case the damping constant of soil is given by equation (2). Following the same consideration as before, the equivalent damping ratio of the structure due to soil is for vertical vibrations in thejth mode

in which generalized mass n

i =1 Mi = C m i wii (27)

and wU = the modal displacement of the footing. Equations (21) and (26) can give an idea about the factors affecting the damping of a structure due to the

geometric damping of the soil. It can be seen that cross-damping can considerably reduce or increase the damping according to the sign of product c5puu#u, that the damping for each vibration mode must be different, and that the damping of the structure must be much smaller than the damping of the footing alone.

Finally, it may be noted that this approach can be used with embedment omitted or with damping coefficients c derived in another way, e.g. by means of the finite element approach. The approach can also be extended to include the hysteretic damping of soil (internal friction). A similar procedure was used by Roesset, Whitman and Dobry,’ who considered hysteretic damping in addition to the viscous damping; however, the often significant effects of the embedment, footing and structure rotatory inertia, the cross damping terms and torsion considered in this paper were not taken into account in Reference 7.

With large eccentricities in the footing base, the cross-damping terms involving torsion should also be included in equation (21), which can be done using cross-~oefficients.~~ The additional terms for equation (21) are obvious.

Accuracy of the approach and application to rigid bodies The accuracy of the approximate modal analysis can be conveniently verified with a rigid body in which

the level of damping is high and the direct analysis is simple and exact. An example is shown in Figure 5, in which the response curves of the sliding component of the coupled

motion are plotted. The excitation was harmonic and the total moment M ( t ) = Q(t) z,+ M&). The response curves were calculated both directly from formulas given in Reference 11, and from equations (21), (24) and


7 .

bl - 10 bl .I 10

6 0, - Q

6 - z5 - 1 2 r o

d - DIRECT ANALYSIS .._.... MODAL ANALYSIS

0 4 - z 0

In 3 -

Q

w 2 '

In

In W f z

In z I

.2 ' .3 .4 .5 .6 .8 LO 1.2 1.6 2.0 DIMENSIONLESS FREOUENCY 0.. wW . O.1

' I

Figure 5. Sliding component of coupled motion of rigid body computed directly and by means of modal analysis (b , = rn/pr& b, = I / f r i , M&) = Ma cos wt, Q ( r ) = Qo cos wt, u ~ , ~ = the first and second natural frequencies, Z, = ro)

(25), based on modal analysis. The damping calculated from equation (21) is 7 per cent for the first mode and 53.6 per cent for the second mode.

It can be seen that the differences between the two approaches are very small and therefore the accuracy of the modal analysis using predetermined damping ratios is quite sufficient.

The error of the modal analysis can be somewhat greater in other cases; it generally grows with damping, increases beyond the first resonance peak and is usually larger in the rocking component than in the translation component of the coupled motion. In the shown example, the error in rocking was up to 5 per cent in the second resonance region.

The differences between the two approaches can be attributed to the fact that the uncoupled equation (24) is quite rigorous only in cases in which the damping can be exactly diagonalized. Otherwise, the generalized co-ordinates are coupled. It appears that the energy consideration leading to equation (21) is equivalent to the omission of the off-diagonal terms in the matrix of the generalized damping [*]T[c][@].

The accuracy of the modal approach, based on (21) for modal damping, to structures was also checked by Ra i r~e r .~~ He compared modal damping computed from (21) with equivalent damping obtained from transfer functions computed directly. He found differences quite acceptable for design analysis.

The good accuracy of the described approach appears rather natural when it is realized that the damping is very important only in the resonant range in which, however, the total response is dominated by the resonating mode as long as the damping is small and the natural frequencies are well separated; and for the resonating mode, the damping was established rather accurately. Actually, the approach seems to work reasonably well even with larger damping if the phase differences between modal components are accounted for.

The modal approach can be further simplified to yield useful approximate formulas for the amplitudes of coupled motion at resonance. The first resonant amplitudes are most important with rigid bodies because the second resonant peak is usually quite suppressed, as the example plotted in Figure 5 indicates. The contribution of the non-resonant modal components to the resonant amplitude can be, in most cases, neglected because it is small and the phase difference between the components is close to 90". Then the resonant amplitudes of the coupled motion at the first resonance are approximately

P1 Ul Pl $1 "= 2D1M1wq' *'= 2D1M1w2,

88 MILOS NOVAK

in which ul, a/t1 and w1 are the first modal displacements and natural frequency obtained from formulas given in Reference 11 and D,, Ml are given by equations (21) and (22).

The resonant amplitude calculated from (28) is also plotted in Figure 5 and is only 0.86 per cent lower than that obtained from the direct analysis; the amplitude of the rocking component is 2.47 per cent. lower. The very simple equations (28) are apparently quite adequate and apply with any number of degrees of freedom for any slightly damped resonance.

Also shown in Figure 5 is a response curve directly calculated with the omission of the cross-damping coefficient cxt. Such an omission can clearly lead to gross underestimation of the response. However, the importance of the cross-damping varies and depends on the position of the reference point. With no embedment and z, = 0, the cross-damping vanishes. z, should be the height of the centroid.

With G, = G and p , = p , the modal damping ratios for coupled motion of a rigid body are independent of shear wave velocity as is the case with one degree of freedom.24 If the stiffness of soil below the base differs from that of the backfill, the modal damping ratios depend on shear wave velocity as indicated by the example (Figure 6).

t 250 500 750 m/sec

2 1 I I I I I I I I

0 1000 2000 3000 ft/sec SHEAR WAVE VELOCITY V,=

Figure 6. Variations in damping ratios of footing from Figure 5 with stiffness of soil: A-stiffness of backfill varies; B-stiffness of backfill does not vary, G, = 108 lb/ft2

THE EFFECT OF SOIL ONDYNAMIC RESPONSE OF STRUCTURES

The outlined approach to modal analysis based on modal damping calculated from equation (21) can be used to illustrate the effect of soil on the dynamic response of structures to external excitation. The consequence of the variations of soil properties is best illustrated using an example. A tall reinforced concrete chimney is examined in this paper as this is a typical structure requiring a dynamic analysis. The dynamic effects to be considered with a chimney are wind gusting, vortex shedding and earthquake.

The chimney is 1,000 ft (304.8 m) tall, the outer diameter is 44.7 ft (13.6 m) at the top and 84.7 ft (25.8 m) at the base. The foundation is a circular flat slab with a radius of 100 ft (30.5 m) and an embedment ratio Z/ro = 0.25. A footing radius of 85 ft (25.9 m) is also assumed for comparison of damping.

Only horizontal translation and rotation in the vertical plane are taken into account. The stiffness of soil is considered variable and is characterized by the shear wave velocity. Soil density p is

considered constant and equal to 3.6 slug/ft3. The lowest shear wave velocities may be considered only of theoretical interest for a chimney of the size considered. The shear wave velocities higher than about 2,000 ft/sec (610 m/sec) correspond to bedrock.

The stiffness and damping parameters for this example are considered invariable, as given in Table I. Variable parameters could also be considered as well if desired; however, for the purpose of considering the overall trends, constant parameters are adequate.


Natural modes, frequencies and modal damping In the first part of the analysis, the natural modes and undamped natural frequencies were calculated with

various soil stiffnesses. The foundation characteristics were included into a computer program originally written by B. Vickery and based on the Stodola method. The effects of shear, rotatory inertia of elements and axial force upon the natural frequencies were found to be negligible. The effect of the rotatory inertia of the footing is appreciable.

Examples of the first four vibration modes computed with two largely different shear wave velocities of soil are shown in Figure 7. It can be seen that the modes change considerably with the stiffness of the soil and that the chimney footing slides and rotates differently according to the soil condition and the order of

/ I

2 3 4 - 7000 ft/sec, ---- 550 ft/sec Figure 7. First four modes of natural vibrations of a 1,OOOft chimney on very stiff and moderately soft foundations.

(Computed with shear wave velocity of soil equal to 7,000 and 550 ft/sec, i.e. 2,134 and 168 m/sec)

the mode. With the first mode or stiff rock essentially only foundation rocking occurs. With soft and moderately stiff soils both sliding and rocking of footing participate in the motion in higher modes. Even the fairly rigid rock does not exclude foundation rocking in the modes of order higher than two.

Associated with the changes in the shape of the modes are the variations of natural frequencies. These are shown in Figure 8 where the first four natural frequencies are plotted vs shear wave velocity of the soil. All the natural frequencies decrease with decreasing stiffness of the soil as expected.

With the natural modes established the damping due to soil is obtained from equation (21), for each particular mode shape. The damping ratios found are also shown in Figure 8. The general trend for the damping is to increase with the order of the mode and with decreasing stiffness of the soil.

Both of these factors have been observed in field experiments. An increase of the damping with the order of the mode was clearly noticeable in experiments analysed by Whitman, Protonotarios and Nelson.28 The interesting theoretical suggestion that the soil damping increases with decreasing quality of soil is supported by in situ measurements reported by K ~ b a y a s h i ~ ~ who experimentally established trends closely resembling those in Figure 8 for the first three modes.

The fourth modal damping indicates, first an increase with decreasing soil stiffness and then a decrease. The other modal dampings exhibit a similar trend if the study is extended to include extremely soft soils. This trend is due to changes in the modal co-ordinates of the footing.

90 MILOS NOVAK

J 30 50L I-

z o m

a - 10 z a a u fi

u a a L L

1.0 '

(3 z a .6 E a .4 n

.2

-'' \ --. '. '.

-. 100 200 306' \ '40Q m/sec

0. I I I I I 1 I -I 0 200 400 600 800 1000 1200 1400ft/sec

SHEAR WAVE V E L O C I T Y O F S O I L Figure 8. Natural frequencies and modal damping ratios of chimney computed with various shear wave velocities of soil

The soil damping is also quite sensitive to other parameters particularly the footing radius and the embedment ratio. To illustrate this point, natural modes and modal damping were also established with a lighter footing having a radius of 85 ft (25.9 m) and the same embedment ratio. The damping variations are shown in Figure 8.

Response to gusting wind The response in the direction of the wind has a mean component independent of the dynamic characteristics

and a fluctuating component dependent both on the dynamic characteristics of the structure and on the nature of the wind. The fluctuating part of the response can be treated as a stationary random process which depends on a number of factors. For this reason, the effect of soil properties cannot be readily seen. This effect can be conveniently examined in terms of the gust effect factor which is the ratio of the expected peak response (load) to the mean response (load) and which reflects the dynamic action of the wind.

A few approaches are available to examine the gust effect factor.=- In this paper Davenport's approach% is applied because it is widely used and because it was adopted in a slightly modified form in the National Building Code of Canada 1970.

The peak factor g, appearing in the gust effect factor, is equal to the ratio of the expected peak value of the displacement to the root-mean-square displacement and is obtained from Davenport's relationship=

in which T is the period of observation taken as 1 hr (3,600 sec) in this example and v is the apparent frequency (number of positive zero crossings) calculated from the spectral density of the displacement.

The modal damping due to soil was added to the constant structural damping of 0.5 per cent to yield the total damping which was used in the analysis.$ The design wind speed of 90 ft/sec (27.4 m/sec) was assumed.

~ ~~

1 The structural component of damping can diminish due to foundation flexibilitye* but is considered to be constant in this example. This seems acceptable for small structural damping.


v U LL

I- V ," 2.5 LL W

I- v) 3 a

2.0

The gust effect factor obtained with different soil properties is shown in Figure 9. Three exposures were considered as indicated because they govern the mean wind profile and the intensity of turbulence.

It can be seen that with decreasing stiffness of the soil the gust effect factor decreases. This trend implies that the poorer soil should benefit the structure by reducing the dynamic component of the response to gusting wind.

A - O P E N E X P O S U R E

8 - S U B U R B A N A R E A S

C - L A R G E C I T I E S rr 2

1

-

-

2 00 30 0 400 m / s e c I I I

1 I I I I I I

S H E A R W A V E V E L O C I T Y OF S O I L Figure 9. Variations of gust effect factor for chimney with shear wave velocity of soil (three exposures)

The dependence of the response on the nature of the terrain is minor because the chimney considered is very tall and reaches very close to the level of the gradient wind where the effect of surface roughness on wind characteristics is not too significant. With lower structures the difference in the effect of the three exposures would be more marked.

Davenport's approach considers just the first vibration mode which is well justified by experimental observations. All vibration modes can be taken into account together with some other refinements in the approach by V i ~ k e r y . ~ ~ However, the trends shown in Figure 9 can be expected to remain the same.

Response to vortex shedding Vortex shedding produces lateral vibrations in the direction perpendicular to that of the wind. These

across-wind oscillations are of particular importance with cylindrical structures such as chimneys and towers because they occur in the broad region of relatively low wind speeds that occur frequently and fatigue failure may result.

The vortex induced oscillations can be treated as a stationary narrow band random process. Vickery's approach36 appears most suitable to examine the effect of vortex shedding from the stack because it is specifically designed for cylindrical structures featuring a variable cross-section (taper). Hence, the response was calculated using this approach and Strouhal number S = 0.22, lift coefficient CL = 0.2, dimensionless correlation length L = 1.0 and the mean wind profile exponent 01 = *. The structural damping added to the damping due to soil was considered as 0.5 per cent and 1 per cent. The peak response was calculated using equation (29) with T = 3,600 sec. The results are plotted in Zgure 10 in the form of relative variations of bending moments with soil properties. Shown are the maximum peak moments for the response in the first mode and the maximum peak moments at a level of three-quarters of chimney height for the response in the second mode. The latter moments are often decisive for design of the upper part of the chimney. However, in this particular case, most of the critical wind velocities for the second mode are above the design wind speed.

92 MILOS NOVAK

- B E N D I N G M O M E N T S

-- H E I G H T O F C R I T I C A L E X C I T A T I O N

C R I T I C A L W I N D SPEED ----

'100 200 300 400 rn/sec

0 200 400 6 00 800 1000 1200 ft/sec 1400 0 I

I I I I I I 1

S H E A R W A V E V E L O C I T Y OF s o i L

Figure 10. Variations of bending moments, relative height of critical excitation and critical wind speed with shear wave velocity of soil for lateral response of chimney due to vortex shedding. (Dstr = structural damping)

The variations of the wind speeds at which the maximum peak response is obtained in the first mode are shown in Figure 10 as well. Also plotted is the relative height (= z / H ) at which the local resonance appears between the Strouhal frequency and the first natural frequency when the maximum response is reached. It can be seen that the critical excitation yielding maximum response does not act at the tip of the chimney as is often assumed. (VickerPs was the first to stress this point.) The critical excitation acts at about three- quarters of chimney height with a rigid foundation and the point of its action moves downward with decreasing stiffness of the soil. The critical wind speed decreases similarly which is significant because the excitation force is proportional to the square of the wind speed. These effects together with the increase in soil damping result in a dramatic reduction of the dynamic response with decreasing stiffness of the soil.

Response to earthquake For the evaluation of the earthquake response a number of approaches are available. The deterministic

approach based on the use of the smoothed pseudo-velocity spectra is used here because it is widely used in practice and because it can indicate smooth trends without any appreciable scatter. Housner's3' smoothed El Centro spectrum reduced to a maximum acceleration of 20 per cent g was used in this paper after numerical extension to very low frequencies and high dampings. The free-field motion was assumed independent of soil properties.

The response in the first three modes was examined for an assumption of structural damping equal to 0-5 per cent which was added to the damping due to soil. The variations of maximum moments with soil properties are plotted in Figure 1 1 . The bending moments are shown in dimensionless form relative to the response of the chimney with a rigid foundation, i.e. with no soil-structure interaction.

It can be seen that the response in the first and third mode decreases as the soil becomes softer, with the decrease in the third mode being very fast. The response in the second mode first increases with decreasing soil stiffness and then rapidly decreases with extremely soft soil. The trend of the first modal response resembles that obtained for a single-storey building by Jennings and Bielak.s The variable trend of the second mode is due to the marked variations of the footing motion and the counteracting effects of the variations of


S H E A R W A V E V E L O C I T Y OF S O I L Figure 1 1 . Variations of bending moments of chimney due to El Centro type earthquake with shear wave velocity of soil

frequency and damping. The rapid decrease in the response with decreasing stiffness is mainly due to the fast increase in soil damping (Figure 8).

Contrary to the behaviour of shear buildings, the true base moments and base shears are of the same order in all three modes in the case of rigid foundation. However, all the relations are very sensitive to the parameters of the structure and its footing.

The beneficial reduction of response due to decreasing soil stiffness may be counteracted by the amplification of the earthquake input typical of soft soils.

CONCLUSIONS

The properties of soil can be conveniently introduced into the analysis of structures and footings in terms of modal superposition. This approach can be facilitated by the determination of soil damping pertinent to each vibration mode.

For any particular structure, the modal characteristics such as natural frequencies, modal displacements and modal damping vary with soil stiffness, dimensions of the footing and embedment ratio. The modal damping due to soil generally increases with the order of the mode and the softness of the soil and can greatly depend on the cross-damping of the footing.

For particular foundation conditions, the soil-structure interaction depends primarily on the stiqness of the structure and on the nature of external excitation. The dependence on the nature of external excitation is illustrated for the example of a tall chimney founded on soil of variable stiffness and exposed to the dynamic effects of gusting wind, vortex shedding and earthquake.

The dynamic response of the chimney to gusting wind decreases by a maximum of about 25 per cent with decreasing soil stiffness.

The across-wind response to vortex shedding decreases very dramatically with decreasing soil stiffness. This reduction is due to decrease in speed and height of the critical wind and to the increase in soil damping.

The response to an El Centro type earthquake decreases with decreasing stiffness in the first and third vibration modes. The second mode shows an initial increase and then a rapid decrease in response. These relations may be different for other types of earthquakes.

The example indicates that with any structure, the soil-structure interaction effects depend on all the factors involved, i.e. soil, footing, structure and the nature of external excitation. In most cases, the general trend of the soil-structure interaction effects is to reduce the response to dynamic loads.

94 MILOS NOVAK

ACKNOWLEDGMENTS

This study was supported by a grant-in-aid of research from the National Research Council of Canada. The assistance of J. Howell and T. Nogami is gratefully acknowledged.

APPENDIX I

Notation The following symbols are used in this paper:

oro J(p/G) (or = o r o J(ps/Gs)) = dimensionless frequency of excitation mlpr; = mass ratio I/pri = inertia ratio elastic half-space stiffness and damping parameters for motion in direction i (l/uo) Ci2 = elastic half-space reduced damping parameter for motion in direction i (may be considered constant contrary to Ciz) constant of damping force in direction i with unit velocity in direction j c/cCrit = damping ratio due to soil pertinent to thejth vibration mode of structure total damping ratio shear modulus of half-space (soil) shear modulus of side layer (backfill) mass moment of inertia of footing about horizontal axis passing through centroid mass moment of inertia of i th lumped mass about horizontal axis mass moment of inertia of faoting about vertical axis Bessel functions of first kind of order 0 and 1 respectively stiffness constant of soil = force (moment) acting in direction i with unit displacement in direction j depth of embedment of footing moment of excitation acting on mass mi excitation moment generalized mass of j th mode amplitude of total excitation moment mass of footing rotating unbalanced mass ith lumped mass generalized force of j th mode horizontal excitation force amplitude of horizontal excitation force amplitude of generalized co-ordinate of mode j generalized co-ordinate of mode j radius of footing base; equivalent radius of footing base side layer stiffness and damping parameters for motion in direction i (l/uo)Siz = side layer reduced damping parameter for motion in direction i (may be considered constant contrary to Siz) moment of excitation in horizontal plane time footing horizontal translation horizontal translation of ith mass horizontal translation of ith mass in j th mode horizontal translation of footing in j t h mode amplitude of total horizontal translation Bessel functions of second kind of order 0 or I respectively height of reference point (CG)above footing base


z, height of horizontal force above reference point (CG) 6 I/r, = embedment ratio p mass density of elastic medium; mass density of undisturbed soil

ps mass density of side layer; mass density of backfill t,h rocking component of footing vibration

t,hi rotation of ith mass fGij rotation of ith mass in modej t,hi rocking of footing in mode j w circular excitation frequency wi j t h circular natural frequency

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