The effect of self-weight and vertical ground acceleration during earthquakes on vertical cantilevers has been studied. The input is taken to be a bivariate normal random process, digitally simulated on a computer. The tip deflection, base moment and shear force have been obtained numerically for three structures of different natural frequencies. It is found that the presence of self-weight and vertical ground excitation could alter these three quantities considerably. This leads to the conclusion that with tall structures a refined analysis, similar to the one presented here, is advisable

Iyengar, R. N.

Shinozuka, M.

English

Indian Academy of Sciences

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 1 ,  69-78 (1972) 
EFFECT OF SELF-WEIGHT AND VERTICAL 
ACCELERATION ON THE BEHAVIOUR OF TALL STRUCTURES 
DURING EARTHQUAKE* 
R. N. IYENGAR-~ AND M. SHINOZUKA~ 
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York 
SUMMARY 
The effect of self-weight and vertical ground acceleration during earthquakes on vertical cantilevers has been 
studied. The input is taken to be a bivariate normal random process, digitally simulated on a computer. The tip 
deflection, base moment and shear force have been obtained numerically for three structures of different natural 
frequencies. It is found that the presence of self-weight and vertical ground excitation could alter these three 
quantities considerably. This leads to the conclusion that with tall structures a refined analysis, similar to the 
one presented here, is advisable. 
INTRODUCTION 
In recent years there has been a steady growth of understanding of earthquakes and their effects on structures. 
In the analysis of tall buildings, chimneys and towers it is customary to replace the structure by a suitable 
model such as a shear beam or a canti1ever.l One then proceeds to obtain the deflection, bending moment 
and shear force under the horizontal ground motion. Though the effect of the horizontal component seems 
to be more important, the effects of vertical acceleration and distributed mass need to be understood. I t  may 
be pointed out here that in buckling analyses of columns free at the top, self-weight is also usually included.2 
In the present paper an effort is made to study these two effects on tall structures which could be idealized 
as uniform cantilevers. The effect of the vertical acceleration is considered only to change the weight of the 
structure. The extensional motion and the effect of the second horizontal component are not included in 
the analysis. 
ANALYSIS 
Referring to Figure 1, the equation of motion of the cantilever can be written as 
where 
U(y,  t )  = absolute lateral deflection 
EZ(y) = flexural rigidity 
m(y) = mass density 
* This work was supported by the National Science Foundation under NSF GK 24925. 
t Presently, Lecturer, Indian Institute of Science, Bangalore-12, India. 
3 Professor 
Received 6 September 1971 
69 
@ 1972 by John Wiley & Sons, Ltd. 
70 R. N. IYENGAR AND M. SHINOZUKA 
c = viscous damping coefficient 
x,, j;, = horizontal and vertical ground acceleration 
Figure 1 .  Cantilever model 
The second term on the right-hand side of equation (1) represents viscous damping assumed to be propor- 
tional to the relative velocity. If the flexural rigidity and mass distribution remain uniform equation (1) can 
be simplified as 
iYV m a 2 V  8VJ m (a2V .) c aV 
-+-(g+jj ,)  (y-h)-+- =-  -+x --- 
ay4 EI [ a p  a,) at2 EI at 
where V is the relative motion with respect to the ground, 
VY, t )  = U(Y, t )  - X g ( 0  (3) 
An approximation in a Galerkin sense (boundary conditions are satisfied but not the differential equation) 
is used to solve equation (2) using the eigenfunctions of a cantilever beam.3 Accordingly, the solution of 
equation (2) is assumed in the form 
N 
n =1 
V(Y, t )  = C. An(t)  @ n b )  
where N is a positive integer, A,(?) are the generalized co-ordinates and cD,(y) are the eigenf~nctions;~ 
On(y) = (cosh @- cos 
h 
where 
A t  = mwi h4/EI 
A, = 1.8751, A, = 4.6941, A3 = 7.8548 
A4 = 10.9955, A, = 14.1371, A, = 17.2787, ... 
Substituting equation (4) into equation (2) and minimizing the mean square error over h one gets 
(4) 
( 5 )  
in which 
BEHAVIOUR OF TALL STRUCTURES DURING EARTHQUAKE 71 
where the prime indicates differentiation with respect to y .  For given inputs ig and j ig  the rlumber N of 
equations in the above system has to be determined weighing the accuracy of the result against the com- 
putational work involved. It is to be noted that the existence of j ; ,  gives rise to a system of equations 
[equation (6)] with time-varying coefficients. 
The ground excitations are random functions. In the l i t e r a t ~ r e ~ - ~  many models have been proposed for 
the horizontal component. At present, however, it appears that not much is known about the vertical 
component and its correlation with the horizontal component. It may be pointed out here that there is no 
reason to believe the two components to be uncorrelated or the statistical structure of j ; ,  to have insignificant 
effects on the solution. In the face of lack of definite information, it is decided in this study to choose a 
bivariate stationary Gaussian process for Zg and pg with the following spectral properties : 
Power spectrum of x,: 
Power spectrum of yg: 
Cross spectrum between 2, and j g :  
The two components are generated using the expressions 
S12(w) = S, exp (- w2/w;) 
n 
z =1 
x,(t) = J(2Aw) C HIl(wi) cos (mi r + Qli) 
where 
HI, = (Sll)+ 
H21 = S12IHll 
A* = -u (w -4 
n 
wi = (i- 4) Aw, W: = wi + SW 
Oli, QZi = uniformly distributed random number in ( 0 , 2 ~ )  
w,, w,, = lower and upper cut-off frequencies in the spectra 
6w = very small random frequency uniformly distributed in - Awl2 and Awl2 
The theory behind the above procedure and other details have been presented e l s e ~ h e r e . ~ ~ ~ ~  It  is expected 
that the structure subjected to such an excitation will produce a typical motion from which a good under- 
standing of the structural response under real seismic excitation can be obtained. 
72 R. N .  IYENGAR AND M. SHINOZUKA 
RESULTS AND DISCUSSION 
The most significant responses of the structure under study would be the relative deflection at the tip, 
bending moment and shear force at the base. In the present case these are given by 
N 
i=l 
N 
i =1 
N 
i=l 
V(h, t )  = x A&) @,(A) 
M(0 ,  t )  = x A&) [Ez(D;(0)+m(g+jjg)gil 
S(0, r )  = A&) EZaq(0) 
These three responses have been studied in some detail for three cantilevers of different natural frequencies 
and heights. These are designated as structures I, I1 and I11 and their properties are as in Table I. For 
Table I 
I 50 10 3.6 x lo8 1.343 
I1 100 10 3.6 x lo8 0.336 
111 200 10 25-2 x lo8 0.222 
* Fundamental frequency. 
purposes of comparison, the responses have been computed also by neglecting the self-weight and vertical 
acceleration. In each case, the system of equations given by equation (6) has been solved numerically by a 
Runge-Kutta-Gill procedure on an IBM 360 computer for various sample inputs. The damping coefficient 
c has been taken as 
c = 2 r p l m  (1 5 )  
where -q = 0.01 and w1 is the fundamental natural frequency of the structure in radians per second. For the 
simulated ground motion the various constants in equations (7-9) have been selected as 
wg = 6i7, p = 0.5 
I s1 = o*02p/7Twg 
s, = O*O5S1/p J S, = 2*25S1/(1 +4p2) 
A suitable choice of the number N of equations to be retained in equation (6) is essential for numerical 
accuracy as well as economy in computer cost. In view of this, equation (6) has also been studied in the 
frequency domain considering 
(17) 
2 = e iAf  
g 
jg = 0 
A .  = 
3 3  
Substitution of this into equation (6) yields a system of algebraic equations for Hi, which could be 
solved easily. Using these values of Hj ,  the frequency response functions of V(h, t ) ,  M(0, t )  and S(0, t )  can 
be obtained as 
BEHAVIOUR OF TALL STRUCTURES DURING EARTHQUAKE 73 
Figure 2 shows the result of such analysis performed on structure I1 plotting the amplitude of the frequency 
response function. The input spectra are also shown in the same figure. The input spectra decay very fast 
i c  
10 
- 
f? 
I < 
i 
I 
10 
10 
1 
-i 
X (rod/sec) 
Figure 2. Frequency response function of structure I1 and input spectra 
and it is seen from the figure that only the first five modes significantly contribute to the response. Accordingly 
in equation (6) five equations ( N  = 5)  have been retained. Since the responses are random, one needs to know 
their statistical properties in as much detail as possible. Herein estimates have been obtained for the r.m.s. 
responses of all the three structures considered. Nine samples for structure I and nineteen samples for the 
other two have been used in arriving at the estimates by ensemble-averaging. Three typical sample responses 
are shown in Figures 3, 4 and 5. Only nine sample responses are computed for structure I since its natural 
frequencies are much higher than those corresponding to other structures and hence it is much more time- 
consuming to perform a time-domain analysis including up to the fifth mode. Figures 6, 7 and 8 show the 
4 
74 R. N.  IYENGAR AND M. SHINOZUKA 
Time (sec) - t. 
Figure 3. Response history of structure I 
-05 - 
-1.OC 
o’8 t A 
-0.8 [- ----Selfweight and neglected 
Time (set)- 
Figure 4. Response history of structure 11 
Time (sec)- 
Figure 5. Response history of structure I11 
BEHAVIOUR OF TALL STRUCTURES DURING EARTHQUAKE 75 
OO r i 2 3 4 5 
08 
0.05 t rJ 
0.6 
0 
0 1 2 3 4 5 - 
G 
b* 0.4 
- 
P 
, G  
- 04 9 
by 
v 
0.2 7' 9 0.2 
Selfweight and & neglected 
I I I I I 0 
1 2 3 4 5 
Time (set)- 
Figure 6 .  R.rn.s. response of structure I 
Selfweight and 4 neglected f ---- 
I 1 I I I 
1 2 3 4 5 
Time (secl- 
Figure 7. R.rn.s. response of structure I1 
estimates of the r.m.s. responses as functions of time. These figures clearly indicate the effect of the self- 
weight and the vertical acceleration. As one could expect, the difference is more considerable with the taller 
structures (11, 111) than with the shorter one. For structural design the highest absolute peak responses are 
more significant and hence these are also obtained in all the samples during the first five seconds of 
earthquake. Since the number of samples is small, it is not possible to obtain a reliable estimate of the proba- 
bility density function. Instead, the (absolute) maximum responses of all the samples have been presented 
in Figures 9-1 1. It is seen that the consideration of self-weight and j ; ,  might either increase or decrease 
the peak responses. However, the difference either way seems to be considerable in most cases. 
76 R. N. IYENGAR AND M. SHINOZUKA 
I I I I I 
0 1 2 3 4 5 
0.6 
- 0.4 4 
bQ 
Y 
P' 
Q 0.2 
Selfweight and $ neglected 
I I I I I 
0 1 2 3 4 5 O Y  
Time (sec) - 
Figure 8. R.m.s. response of structure I11 
0 
d 
0 
-. I 
1 3 5 7 9 
( a )  
( b )  
f o r  o 
' 8  
e d 
t l  , 
5 7 9 3 1 
Sample number 
( C )  
Self-weight and Y,: Considered 0, Neglected 0 
Figure 9. Effect of self-weight and vertical acceleration 
for structure I in terms of absolute maximum response 
BEHAVIOUR OF TALL STRUCTURES DURING EARTHQUAKE 77 
7.n L 4.28 0 3-92 a a 0 
Q 
0.6 
c 0 r Q 
0 
I .  
1 5 10 15 
2.0 c 
a 
0 
0 
(3 
Sample number Sample number 
( C )  (C) 
Self-weight and Y g :  Considered 0, Neglected 0 
Figure 10. Effect of self-weight and vertical acceleration 
for structure 11 in terms of absolute maximum response 
Figure 11.  Effect of self-weight and vertical acceleration 
for structure 111 in terms of absolute maximum response 
The above result implies that the effect of vertical acceleration could be much more pronounced, particu- 
larly in beam response, when a frame structure is considered without recourse to the shear building 
assumption. A general method of dynamic response analysis in which the distributed mass of the frame 
structure can be taken into consideration was developed in Reference 11. It will be an interesting future 
study to assess the effect of vertical acceleration on the response of the frame structure by combining the 
method in Reference 11 with the simulation technique described here. 
78 R. N. IYENGAR AND M. SHINOZUKA 
REFERENCES 
1. R. W. Clough, Earthquake Engineering (Ed. R. L. Wiegel), Prentice-Hall, Englewood Cliffs, New Jersey, 1970, pp. 
2. S. Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York and London, 1936, p. 115. 
3. Handbook of Engineering Mechanics (Ed.) W. Fliigge, McGraw-Hill, New York, Toronto, London, 1962, pp. 61-8-61-9. 
4. M. Amin and A. H.3 .  Ang, ‘Nonstationary stochastic models of earthquake motions’, EMD J. Am. SOC. Ciu. Engrs, 
5. G .  W. Housner and P. C. Jennings, ‘Generation of artificial earthquakes’, EMD J., Am. SOC. Ciu. Engrs, 90, 113-150 
6. M. Shinozuka and Y. Sato, ‘Simulation of nonstationary random process’, EMD J. Am. SOC. Ciu. Engrs, 93,1140 (1967). 
7. J. E. Goldberg, J. L. Bogdanoff and D. R. Sharpe, ‘The response of a simple nonlinear system to a random disturbance 
of the earthquake type’, Bull. Seism. SOC. Am. 54, 264-276 (1964). 
8. R. N. Iyengar and K. T. S. Iyengar, ‘A nonstationary random process model for earthquake accelerograms’, Bull. 
Seism. SOC. Am. 59, 1163-1188 (1969). 
9. M. Shinozuka, ‘Simulation of multivariate and multidimensional random processes’, J. Acoust. SOC. Am. 49, 357-368 
(1971). 
10. M. Shinozuka and C.-M. Jan, Simulation of Multivariate and Multidimensional Processes 11, NSF G K  3858124925 
Technical Report No. 12, Department of Civil Engineering and Engineering Mechanics, Columbia University, April 
1971. 
307-334. 
94, 559-583 (1968). 
( I  964). 
11 .  M. Shinozuka and J. N. Yang, ‘Random vibration of linear structures‘, fnt. J. Solids Struct. 5, 1005-1036 (1969). 


Effect of self-weight and vertical acceleration on the behaviour of tall structures during earthquake

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Effect of self-weight and vertical acceleration on the behaviour of tall structures during earthquake

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