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Effect of vertical support on highway bridge superstructures
004s7949/91 s3.00 + 0.00
Computers d Structures Vol.40,No.4, pp. 947-955. 1991 Printed in Gnat Britain.
EFFECT
0 1991Pergamon Press plc
OF VERTICAL SUPPORT ON HIGHWAY SUPERSTRUCTURES
BRIDGE
J. S. DAVIDSON and C. H. Yoo Department of Civil Engineering, Auburn University, Auburn, AL 36849, U.S.A. (Received 7 June 1990) Abstract-A Fourier series expansion is used to obtain the deflection, moment, shear, velocity, and acceleration expressions for a multispan highway bridge superstructure under the effects of vertical support motion. These expressions are used in computer programs that compute the maximum values along the bridge span. The formulation is also used to show the feasibility of creating nondimensional&d design aids that may be used in industry. A finite element analysis is used to help verify the results.
2. FORMULATION OF THE PROBLEM
1. INTRODUCTION
is one of the most prevalent and destructive forms of dynamic disturbances in the United States. Contrary to the commonly held belief that the California coast is the only area subject to strong seismic disturbances, the United States Coast and Geodetic Survey (USCGS) records show that these destructive forces have been felt all over the country [ 11. Up until the end of the nineteenth century, little if any, attention was given to earthquake forces in structural design. After the great California earthquake of 18 April 1906, the high population concentration of the twentieth century cities emphasized the importance of earthquake design and the development of displacement meters and accelerometers capable of recording earthquake movement encouraged earthquake studies. A number of these studies suggest that punching shear-type failure at the supports is prevalent in bridge-type structures yet the America1 highway and railroad bridge design codes (AASHTO and AREA) do not specify any provisions against the vertical effect of earthquakes on bridges [2, 31. In the following sections, the dynamic response of highway bridge superstructures to pulsating support settlements induced by earthquake excitation including the effects of damping is presented. To conduct the investigation, a differential equation governing the motion of a multispan bridge due to pulsating support settlements induced by an earthquake was derived and the deflection, moment, shear, velocity, and acceleration expressions were obtained in terms of a Fourier series expansion. A computer program was developed to calculate these expressions. The parameters in these expressions were then nondimensionalized so that design charts could be developed. In Section 3 a two-span bridge example was used to demonstrate the use of the computer program and the design charts. A finite element analysis is used to verify the results. The earthquake
947
General discussion
A highway bridge superstructure may be subjected to additional stresses from the vertical excitations of an earthquake, arising from: (1) the inertial forces acting on the bridge as a result of the rigid body translation, and (2) the effect of dynamic differential settlement of one or more piers. The stresses due to the inertial force can be obtained by assuming that the bridge is subjected to an additional static-equivalent uniformly distributed load of a magnitude ma, where m is the mass per unit length of the bridge superstructure and a is the acceleration of the rigid body translation. As discussed earlier, the additional stresses due to the dynamic differential support settlements require further analysis. Presented in this section is a mathematical formulation to determine the additional stresses and deflections resulting from these settlements. Assumptions used in the Fourier series formulation
In order to determine the response of the structure due to seismic loading, a set of simplifying assumptions is made to make the problem less complex. The assumptions adopted in the Fourier series formulation are listed below. Reference to the validity of these assumptions is given in the derivation process whenever it is appropriate. 1. The bridge superstructure is analyzed as a twodimensional structure. 2. The slab sections are considered to be pinned at the abutments. 3. The elastic properties are assumed constant along the length of the bridge. 4. The mass is assumed uniformly distributed and constant along the length of the bridge. 5. The supports are assumed to vibrate in a harmonic motion in the vertical direction under the effect of the earthquake.
J. S. DAVIDSONand C. H. Yoo
948
6. The system is considered to behave within the limits of the elastic range. 7. The effect of axial deformation, if any, is ignored. 8. The effect of shear deformation is ignored. 9. The effect of rotary inertia is neglected. Fourier series formulation
A multispan highway bridge superstructure can be modeled by a wide continuous beam, with pinned ends and intermediate supports. In order to determine the maximum deflection, moment, shear, velocity, and acceleration, at any time, for any point along the span, the following steps are taken to determine these values: 1. The intermediate supports are removed, leaving an ordinary simple beam for which the differential equation, vibration frequencies, and normalized shape functions are known. 2. Constraints equivalent to the removed supports are imposed on the simple beam. These constraints are first transformed into concentrated loads on the simple beam, and then substituted by a generalized Fourier series in terms of the normalized shape functions of the simple beam. 3. The deflection of the simple beam at any point along its span is also represented as a generalized Fourier series in terms of the normalized shape functions of the beam. 4. Using the constraints at the intermediate supports, the forced deflection function of the beam is determined. 5. Imposing constraints that allow no deflection at the support locations, the deflection function of the continuous beam for free vibration is determined. 6. Moment, shear, velocity, and acceleration functions can be determined for the continuous beam by successive differentiation of the deflection function with respect to longitudinal coordinate along the span or with respect to time. 7. Numerical values and maximum values for these functions are determined by the use of computer programs. A similar procedure was used by Rogers [4] to investigate the behavior of a continuous beam subjected to a series of pulsating loads. Saibel and D’Appolonia [5] studied similar problems based on the energy of the system and Lagrangian constants. Considering assumptions (6)-(9), the differential equation governing the vibration behavior of a simple
n f I?
f=ll&aca~
v
beam subjected to viscous damping and the lateral forces, shown in Fig. 1, can be written as EI$+m$+C~=w(x,t)
(1)
in which E is the modulus of elasticity, I is the moment of inertia of the beam cross section, m is the mass per unit length, c is the coefficient of damping, and w(x, t) is the distributed lateral load. The natural frequencies of the simple beam are given by
where p,, is the natural frequency of the simple beam vibrating in the n th mode, and L is the span length of the simple beam. The normalized shape functions of the simple beam are readily obtainable, and can be expressed as [4,6,71 . mcx smL
2
Xn(x)=
x
J(
>
in which X,,(x) is the normalized shape function when the simple beam is vibrating in the n th mode. The deflection of the simple beam as a function of the longitudinal coordinate and time can be expressed at [4,61
Y(X,t) = z‘,
\i( > --$
sinyq.(t)
(4)
in which q,,(t) is as function of time only. The constraints imposed by pulsating supports due to vertical earthquake excitations can be idealized as harmonic motions of the form y(xi, t) = Ai sin(+
- c(,)
(5)
in which xi is the position of support i along the span, Ai is the amplitude of the ith support settlement, o, is the circular forcing frequency, and a, is the phase angle between the force and the response due to damping. The force at support i for the n th mode of vibration can be written Fh(t) = R, sin w,t,
(
at2 at Fig. 1. General loading and free body diagram.
Mdbc ax v4Kix ax
(6)
949
Vertical support vibration on bridges
where R, is the undetermined amplitude of the equivalent force for support i. The Fourier series expression for F-,(r) is given as F,(l) = zc, A, x,(x),
(7)
where A, is a constant for each mode of vibration and is such that the series will correctly represent F,(t) along the span. Assuming that the concentrated load acts over an infinitely small distance ci as shown in Fig. 2, and multiplying both sides of eqn (7) by mX,(x) in order to use orthogonality relationships we can solve for the A,s
i;.(t)+~q.(t)fp~g,(t)=~,RiX.(x,)sino,(t).
Assuming the solution to eqn (14) to be of the form q.(r) = D,, sin o/r + D,, cos w,f
X,+ c/2 mC, A, -W)&(x) x,-c/2
I
b.
(8)
Y,k
I
x, +
c/2 mFin(r)Xn (x) dx. Xi- e/2
For a beam with multiple intermediate eqivalent load becomes
(9)
(10)
0 = WXxM),
= ZiZnmFi,,(r)Xn(xi)S,,(x), (13)
Fig. 2. Simulation of a concentrated load. CA9 40,4-J
-I1’
/cl
G -m~,~~~~,f
1
. (16)
form of this solution would be
y(x, r) = Y’ sin(o+r - a,)
(17)
where Y’ is the forced amplitude and a, is the phase angle
and
(12)
where q”(r) is a function of time only for the n th mode of vibration. Substituting the value of w(x, t) given in eqn (11) into (1) yields + C, E + m 2
(p;5-wj)sinw#
-,
(11)
The deflection of the continuous beam can also be expressed in terms of the normalized shape function
Hz
(15)
supports the
x(x, r) = CiC,mFin(r)Xn(xi)Xn(x).
Y(-T
x
A more convenient
In the limiting case when c, approaches zero A, = mFh(r)Xn(xi).
W&(x,)
t) = LX(x)
7
For a beam with no energy dissipation at its supports and with a constant rigidity along its span, the shape functions are orthogonal and eqn (8) reduces to A, =
(14)
and substituting this solution and its derivatives back into eqn (15) and then equating sine and cosine terms yields the total particular solution
mF, (r )Xk (x) dx
=
where C,, is used since damping constitutes a nonconservative force system and thus is assumed to be related to the mode of vibration. After some algebraic manipulation using eqns (12), (2), and (3), (13) becomes
~~~
Yp(x, r) = Gf,(x)
x sin&r
- a,)
(20)
J. S. DAVIDSONand C. H. YCMJ
950
which is a system of linear equations that has as many unknowns (&s) as there are interior supports. Because of the complex nature of damping, there are a number of ways to treat the damping coefficient in this problem. But for this development it is assumed that C, remains proportional to pn so that Cn/pn = constant and the ratio to critical damping e, is a constant t.=t
=& n
c”=2p,5. M
(22)
Using this approach to damping and rewriting eqn (20) yields
&RJ”(Xi)
Y,(x, t) = zJ,(x) [
J[
(Pi - w;j2+ (2P”59)*
11
x sin(w,r - a,)
(23)
which is the solution used in the computer program developments. Although the free vibration or homogeneous solution can be determined by the same method used in the forced vibration method, the deflection due to free vibration cannot be taken into consideration in earthquake problems. This is because the initial conditions imposed by these settlements are not exactly known and therefore the amplitudes of the free vibrations cannot be solved. Saibel and D’Appalonia considered multispan beams with pulsating sinusoidal loads acting within the span [5]. In this case, taking the total initial velocity and total initial displacement equal to zero is a reasonable assumption and would hold true for every point along the span, including the support locations. Using these initial conditions, the shape function X,(x) can be factored out and the total solution can be solved. But in the case of multispan beams with forced support settlements induced by vertical earthquake excitations, the motion at the intermediate supports cannot be said to have zero initial forced velocity. Therefore, the total initial velocity cannot be taken as zero for every point along the span. This implies that X,,(x) cannot be factored out as was done in the sinusoidal load case and thus separation of the time and shape functions is not possible for earthquake initial conditions when using Fourier Series derivation. Another limitation of the Fourier Series derivation is that whenever magnitudes of the reactions Ris are equal to zero simultaneously, determining the natural frequencies by setting the determinant of the reaction coefficients to zero is not possible. This is actually the case of the trivial solution to eqn (21) and is not a major drawback because for beams of constant elastic properties, the intermediate reactions will all be zero only in the case of equal span beams. For these cases the frequencies are equal to multiples of
the simple beam frequencies, and can thus be readily determined. 3. NUMERICAL ANALYSIS
Using the formulation presented in Section 2, a computer program has been developed to calculate the deflection, velocity, acceleration, moment, and shear along the span, as well as natural frequencies of the bridge. The program uses eqn (21) to determine the equivalent support settlement reactions induced by the earthquake and eqn (20) and its derivatives to solve for the various functions. Also from this development, design aids can be produced that may be used in industry to design for seismic disturbances. The dynamic amplification factors are computed as the ratios of the maximum dynamic values (deflection, moment, and shear) to the corresponding maximum static values. These charts can be used for bridges that have any combination of E, I, m, and L, but maintain the same relative ratios of intermediate support distances. In order for the design aids to be usable they must be in a nondimensionalized form. The forcing frequency can be expressed as
a’ = &EIymL’)
The dynamic amplification charts are plotted using wi so that they can be used for any bridge that has the same relative support locations but have any combination of E, I, m, and L. The plots were developed by taking the longest intermediate span length to be 1.0 so that all plots could be produced for the same general range of w;. Therefore, in order to use this set of DAF plots, o; given in eqn (24) must be divided by a factor of TL* where TL is the sum of the span ratios given at the top of each DAF plot
w’ = TL2 J(EI/mL4) Damping
In the Fourier series formulation, it was assumed that the damping coefficient varied with the mode of vibration (viscous damping). The ratio to critical damping (5) was defined and used in the computer program developments. But since the Fourier series formulation is based entirely on the vibration modes ,, of the simple beam, the value input for ZETA may not be representative of the actual damping ratio for the multispan beam. The fundamental frequency of the simple beam is always less than the fundamental frequency of the same beam with intermediate supports so in order to get a more representative input value for ZETA the following relation is used
Vertical support vibration on bridges
951
where (27) psimplcis the fundamental frequency of the simple is the fundamental frequency of the b=m, ~~~~~~~~ multispan beam, < is the ratio to critical damping of the multispan beam previously defined (specified at
the top of each DAF plot as ZETA’), and tinputis the ZETA value used int he program input to achieve a more representative response for damping. 4. PARAMETER
-2 i
LezJth 7)
UOsm6)0M
Fig. 3. The maximum dynamic deflection along the span for a support settlement of 0.0805 inches. 200 terns summed.
STUDIES
Example
To illustrate the use of computer programs, consider a two-span concrete bridge of total length 60 ft and width 32 ft. One span of this bridge has a length of 33 ft and the other a length of 27 ft. The bridge has a haunch at the intermediate support. Although the actual bridge will behave in a three-dimensional way, it seems appropriate to model it as a two-dimensional structure using equivalent cross-sectional properties. These properties are [8]: moment of intertia, 92,850 in’.; depth, 17 in.; Young’s modulus, 3,000,OOOpsi; and mass per unit length, 1.467 lb-sec/in2. This bridge is analyzed under the effects of three forcing frequencies and the effects of two damping cases are demonstrated. Although the acceleration rather than the earthquake frequency is usually documented for earthquakes [9], the forcing frequency in this mathematical model can be readily derived by estimating the support settlements under the effect of acceleration. With both the acceleration and support settlement known, the forcing frequency can be determined by CD,=
a
J(> xi
25 33 33
96.6 128.8 128.8
1w
140
uo
400
‘aa
Length(in)
so
040
720
Fig. 4. The maximum dynamic moment along the span for a support settlement of 0.0805 inches. 200 terms summed.
-40
-so
Table 1. Forcing frequencies used in example problems a, ia./se?
a0
--Y)
in which o, is the forcing frequency, a is the acceleration due to the earthquake, Ai is the expected support settlement under the effect of the earthquake acceleration. These parameters are given in Table 1. The dynamic response to these frequencies can be seen in Figs 3-9. From these figures it can be observed that the deflected shape varies drastically with the variation of the forcing frequency, and that for high frequencies the bridge superstructure exhibits more than one bend along its span. With such deflected shapes, the curvature is sharper and thus higher moments and shear.
a, %g
0
A,, in.
o,, rad/sec
.2415 .0805 .0057
20 40 150
lea
7.40
310
400
4ao
MO
MO
720
Length (in)
Fig. 5. The maximum dynamic shear along the span for a support settlement of 0.0805 inches. 200 terms summed.
1::
Length C)
4ao~s4072a
Fig. 6. The maximum dynamic deflection along the span for a support settlement of 0.2415 inches. 200 tenas summed.
J. S. DAVIDSON and C. H. Yoo
952
-10
.
b
.
.
.
m
.
.
.
. . (80
.
.
. 24a
..~I...........I...I~~~ 320 4m
Length
wo
4m
no
Ma
(in)
Fig. 7. The maximum dynamic deflection along the span for a support settlement of 0.0057 inches. 200 terms summed.
-0.7.
6
.
do
lb
.
.
.
. 20
.
.
.
.
.
.
320
Length
.
.
.
.
.
400
.
.
..I.......
sso
4ao
(in)
no
Ma
Fig. 8. The maximum dynamic deflection along the spin for a support settlement of 0.0805 inches, 200 terms summed.
Fig.
10. Dynamic
b
do
#.
?@a
I..
240
8..
am
8.
400
Length(in)
..
I
4ao
”
I
so
“‘I
”
MO
‘110
Fig. 9. The maximum dynamic deflection along the span for a support settlement of 0.0805 inches. 200 terms summed.
For a forcing frequency of 40 rad/sec, the maximum deflection occurs at 540 in. (45 ft) from the left abutment; the maximum moment at 568.8in. (47.4 ft); and the maximum shear occurs at 410.4 in. (34.2 ft). The fundamental natural frequency of this bridge is computed as 31.43 rad/sec. Now using this same example to illustrate the use of the dynamic amplification charts, the maximum static deflection is 0.081045 in., the maximum static moment is 524300 lb-in., and the maximum static shear is 1618.4. The value of wi for an earthquake frequency of 40 rad/sec can be obtained from eqn (25) 40
fBj= 1.8182’
3,000,000 x 92,850 1.467 (720)4 >
factors
for deflection
Entering the deflection chart Fig. 10 with ZETA = 0 (no damping), the dynamic amplification factor is 2.15 and thus the maximum dynamic deflection expected is 2.15 x 0.081045 = 1.742 in. Other DAF for deflections are given in Figs 11 and 12 for nonzero damping cases. The exact value as produced by the Fourier series program is 1.738 in. The DAF for the moment can be obtained from Fig. 13 as 7.0 and hence the maximum dynamic moment is 7.0 x 524,300= 3,670,100 lb-in. The exact value is 3,649,300 lb-in. The dynamic amplification factor for shear is 36.0 as can be read from Fig. 14 and the maximum dynamic shear is therefore 36 x 1618.2 = 58255 Ibs. The exact value is 55457 Ibs. 5.
-0.7.
amplification functions.
DISCUSSION
OF RESULTS
From the example given in Section 4, and from other examples considered, a good correlation was observed in comparing the results of the DYNAMIC computer program to the results achieved by using the dynamic amplification method. Also, for cases in which damping forces are neglected, it has been shown that a very good correlation exists between the results obtained from using a three-moment equation development and the results obtained from using the Fourier series [lo]. It was observed, however, that more terms have to be summed as the number of bridge spans is increased. For example, it was observed that the summation of the first 50 terms gave reliable results for the frequency computations
Q
= 14.39. Fig.
11. Dynamic
amplification factors functions.
for deflection
953
Vertical support vibration on bridges
Fig.
12. Dynamic
amplification functions.
factors
for deflection
moment amplification, and a substantial increase in the shear amplification values. This is primarily because the shear is related to the rate of curvature which is increased with the higher forcing frequencies. A tinite element analysis was performed on the example presented in Section 4 using ADINA. A modal analysis was used to verify the natural frequencies produced by the Fourier series program and direct integration was used to produce the maximum absolute values of displacement, moment, and shear along the span for the three forcing frequencies used in Section 4 (see Figs 15-19). In general, a good correlation was noted between the deflected shape of
(........................... ........
-0.4
0
a0
180
ma
310
Length
Fig.
13. Dynamic
amplification functions.
factors
for
moment
400
am
I . . . . . . . . . . . . . . . . . . . . . . . . no Ia0 a40 ?no 400 4aa Lmgth (in)
0
as well as for the deflection, moment, and shear functions for two span bridges, whereas 100 and 200 terms are needed to achieve the same accuracy for three- and four-span bridges, respectively. As expected, the convergence of the deflection function is the most rapid due to the existence of the n4 term in the denominator of the deflection expression. The convergence of the moment function is faster than that of the shear function because of the nz term in the denominator of the moment expression as opposed to the n term in the shear expression. In the DAF plots, damping causes a sharp peak at resonance locations with some peaks higher than others. Neglecting the resonance effects, the DAF plots show little increase in deflection amplification with increase in forcing frequency, some increase in
w
no
Fig. 15. The maximum absolute dynamic deflection along span for a support settlement bf 0.0805 inches. Using ADINA.
-10.0
Fig. 14. Dynamic amplification factors for shear functions.
MO
(ill)
..,...,...i MO
640
no
Fig. 16. The maximum absolute dynamic moment along the span for a support settlement of 0.0805 inches. Using ADINA.
--3oo --4oo
1! . . . . . . . . . . . . . . . . . . . . . . . .._... 0
a0
1e4
240
w Length
400
4m
am
_.,_ Ma
no
(In)
Fig. 17. The maximum absolute dynamic shear along the span for a support settlement of 0.0805 inches. Using ADINA.
J. S. DAVIDSON and C. H. Yoo
954
Fig. 18. ine maximum absolute dynamic deflection along the span for a support settlement of 0.2415 inches. Using ADINA.
--P),...,...,...,...,...,...,...,...,... 0
aa
100
240
320
Length
400 (in)
4m
SW
e4a
no
Fig. 19. The maximum absolute dynamic deflection along the span for a support settlement of 0.0057 inches. Using ADINA. the structures and magnitudes of the displacement, moment, and shear. The model used in ADINA produced higher values of deflection, moment, and shear, but within the same order of magnitude. 6. SUMMARY AND CONCLUDING REMARKS The response of highway bridge type structures to dynamic differential support settlement was investigated. The bridge was modeled as a two-dimensional continuous beam with pinned ends and intermediate supports. The governing differential equation of motion (including damping) was derived for pulsating support settlements induced by the vertical acceleration of an earthquake. The deflection, moment, shear, velocity, and acceleration expressions were determined in terms of a Fourier series expression and a computer program was developed to calculate these values. The results of this development were compared to the results of a finite element analysis (ADINA) and a good correlation was noted between deflected shapes and the absolute maximum displacement, moment, and shear values along the span. The parameters in the Fourier series expression were nondimensionalized and plots were created for dynamic amplification factors versus forcing frequency. A two span bridge example was used to illustrate the use of these plots as design aids and a good
correlation was noted between the use of the design aids and the results produced by the Fourier series program. Further investigation should be done on the use of damping in this development. Presently only viscous damping is considered and it is assumed to be proportional to the simple beam frequencies. The peaks in the dynamic amplification plots, indicating dynamic resonances, may not be feasible since considerably dynamic oscillation (duration of vibration) is normally required to develop a full resonance effect. Also, actual earthquake motion is by no means a simple harmonic motion. A study correlating the validity of using simple harmonic motion to represent the structural behavior of the system subjected to an earthquake may be of value. Past performances of bridges during earthquakes such as the San Fernando earthquake [l l-131 suggest that a majority of the bridges which collapsed were subjected to unusually high shear. This investigation verified those observations. As can be seen in the charts, the dynamic amplification factors for shear are quite high, especially under higher forcing frequencies. Until the adoption of more rigid design criteria, a few design hints based on some general observations in this study might be appropriate. Since the dynamic amplification factors for shear were the highest, and probably controlling for all ranges of forcing frequencies, strict attention should be given to the details of the shear resisting members. Also, since the natural frequency is inversely proportional to the second power of the total length of the bridge, bridges with shorter span lengths should be considered in areas of high seismic activity. This increases the natural frequency of the bridge and decreases the dynamic amplification factors. Acknowledgements-Support of this work was partially provided by the National Science Foundation through Grant No. PFR-7822845. The support of Dr John B. Scalzi of NSF is gratefully acknowledged. REFERENCES
1. USCGS, United States Earthquakes of 1956. U.S. Government Printing Office, Washington, DC (1958). 2. M. Chen and J. Penzien, An investigation of the effectiveness of existing bridge design methodology in providing adequate structural resistance to seismic disturbance. Phase III, Report No. FHWA-RD-75-10, University of California, Berkeley (1974). 3. W. Tsen and J. Penxien, An investigation of the effectiveness of existing bridge design methodology in providing adequate structural resistance to seismic disturbance. Phase II, Report No. FHWA-RD-74-3, University of California, Berkeley (1974). 4. G. L. Rogers, Dynamics of Framed Structures. John Wiley, New York (1959). 5. E. Saibel and E. D’Appolonia, Forced vibrations of continuous beams. Trans. ASCE 117,1075-1090 (1952). 6. M. M. Biggs, Introduction to Structural Dynamics. McGraw-Hill, New York (1964). 7. W. Nowacki, Dynamics of Elastic Structures. John Wiley, New York (1973).
Vertical support vibration on bridges 8. R. J. Banerian, An analysis of the behavior of short two span and three span highway bridges subject to pulsating support settlements. Thesis presented to Marquette University, at Milwaukee, Wisconsin, in partial fulfilment of the requirements for the degree of Master of Science (1977). 9. R. L. Wiegel, Earfhquuke Engineering. Prentice-Hall, Englewood Cliffs, NJ (1970). 10. S. V. Acra, Effect of vertical vibration of the supports of a highway bridge on a superstructure. Thesis presented to Marquette University, at Mil-
955
waukee, Wisconsin, in partial fulfilment of the requirements for the degree of Master of Science (1980). 11. A. L. Elliott, Hindsight and foresight on the performante of prestressed concrete bridges in the San Fernando earthquake. PCI JI 17, (1972). 12. H. S. Lew et al., Engineering aspects of the 1971 San Fernando earthquake. BSS40, NBS, Washington, DC (1971). 13. W. F. Pond, Performance on bridges during San Fernando earthquake. PCI JI 17, No. 4 (1972).
Computers d Structures Vol.40,No.4, pp. 947-955. 1991 Printed in Gnat Britain.
EFFECT
0 1991Pergamon Press plc
OF VERTICAL SUPPORT ON HIGHWAY SUPERSTRUCTURES
BRIDGE
J. S. DAVIDSON and C. H. Yoo Department of Civil Engineering, Auburn University, Auburn, AL 36849, U.S.A. (Received 7 June 1990) Abstract-A Fourier series expansion is used to obtain the deflection, moment, shear, velocity, and acceleration expressions for a multispan highway bridge superstructure under the effects of vertical support motion. These expressions are used in computer programs that compute the maximum values along the bridge span. The formulation is also used to show the feasibility of creating nondimensional&d design aids that may be used in industry. A finite element analysis is used to help verify the results.
2. FORMULATION OF THE PROBLEM
1. INTRODUCTION
is one of the most prevalent and destructive forms of dynamic disturbances in the United States. Contrary to the commonly held belief that the California coast is the only area subject to strong seismic disturbances, the United States Coast and Geodetic Survey (USCGS) records show that these destructive forces have been felt all over the country [ 11. Up until the end of the nineteenth century, little if any, attention was given to earthquake forces in structural design. After the great California earthquake of 18 April 1906, the high population concentration of the twentieth century cities emphasized the importance of earthquake design and the development of displacement meters and accelerometers capable of recording earthquake movement encouraged earthquake studies. A number of these studies suggest that punching shear-type failure at the supports is prevalent in bridge-type structures yet the America1 highway and railroad bridge design codes (AASHTO and AREA) do not specify any provisions against the vertical effect of earthquakes on bridges [2, 31. In the following sections, the dynamic response of highway bridge superstructures to pulsating support settlements induced by earthquake excitation including the effects of damping is presented. To conduct the investigation, a differential equation governing the motion of a multispan bridge due to pulsating support settlements induced by an earthquake was derived and the deflection, moment, shear, velocity, and acceleration expressions were obtained in terms of a Fourier series expansion. A computer program was developed to calculate these expressions. The parameters in these expressions were then nondimensionalized so that design charts could be developed. In Section 3 a two-span bridge example was used to demonstrate the use of the computer program and the design charts. A finite element analysis is used to verify the results. The earthquake
947
General discussion
A highway bridge superstructure may be subjected to additional stresses from the vertical excitations of an earthquake, arising from: (1) the inertial forces acting on the bridge as a result of the rigid body translation, and (2) the effect of dynamic differential settlement of one or more piers. The stresses due to the inertial force can be obtained by assuming that the bridge is subjected to an additional static-equivalent uniformly distributed load of a magnitude ma, where m is the mass per unit length of the bridge superstructure and a is the acceleration of the rigid body translation. As discussed earlier, the additional stresses due to the dynamic differential support settlements require further analysis. Presented in this section is a mathematical formulation to determine the additional stresses and deflections resulting from these settlements. Assumptions used in the Fourier series formulation
In order to determine the response of the structure due to seismic loading, a set of simplifying assumptions is made to make the problem less complex. The assumptions adopted in the Fourier series formulation are listed below. Reference to the validity of these assumptions is given in the derivation process whenever it is appropriate. 1. The bridge superstructure is analyzed as a twodimensional structure. 2. The slab sections are considered to be pinned at the abutments. 3. The elastic properties are assumed constant along the length of the bridge. 4. The mass is assumed uniformly distributed and constant along the length of the bridge. 5. The supports are assumed to vibrate in a harmonic motion in the vertical direction under the effect of the earthquake.
J. S. DAVIDSONand C. H. Yoo
948
6. The system is considered to behave within the limits of the elastic range. 7. The effect of axial deformation, if any, is ignored. 8. The effect of shear deformation is ignored. 9. The effect of rotary inertia is neglected. Fourier series formulation
A multispan highway bridge superstructure can be modeled by a wide continuous beam, with pinned ends and intermediate supports. In order to determine the maximum deflection, moment, shear, velocity, and acceleration, at any time, for any point along the span, the following steps are taken to determine these values: 1. The intermediate supports are removed, leaving an ordinary simple beam for which the differential equation, vibration frequencies, and normalized shape functions are known. 2. Constraints equivalent to the removed supports are imposed on the simple beam. These constraints are first transformed into concentrated loads on the simple beam, and then substituted by a generalized Fourier series in terms of the normalized shape functions of the simple beam. 3. The deflection of the simple beam at any point along its span is also represented as a generalized Fourier series in terms of the normalized shape functions of the beam. 4. Using the constraints at the intermediate supports, the forced deflection function of the beam is determined. 5. Imposing constraints that allow no deflection at the support locations, the deflection function of the continuous beam for free vibration is determined. 6. Moment, shear, velocity, and acceleration functions can be determined for the continuous beam by successive differentiation of the deflection function with respect to longitudinal coordinate along the span or with respect to time. 7. Numerical values and maximum values for these functions are determined by the use of computer programs. A similar procedure was used by Rogers [4] to investigate the behavior of a continuous beam subjected to a series of pulsating loads. Saibel and D’Appolonia [5] studied similar problems based on the energy of the system and Lagrangian constants. Considering assumptions (6)-(9), the differential equation governing the vibration behavior of a simple
n f I?
f=ll&aca~
v
beam subjected to viscous damping and the lateral forces, shown in Fig. 1, can be written as EI$+m$+C~=w(x,t)
(1)
in which E is the modulus of elasticity, I is the moment of inertia of the beam cross section, m is the mass per unit length, c is the coefficient of damping, and w(x, t) is the distributed lateral load. The natural frequencies of the simple beam are given by
where p,, is the natural frequency of the simple beam vibrating in the n th mode, and L is the span length of the simple beam. The normalized shape functions of the simple beam are readily obtainable, and can be expressed as [4,6,71 . mcx smL
2
Xn(x)=
x
J(
>
in which X,,(x) is the normalized shape function when the simple beam is vibrating in the n th mode. The deflection of the simple beam as a function of the longitudinal coordinate and time can be expressed at [4,61
Y(X,t) = z‘,
\i( > --$
sinyq.(t)
(4)
in which q,,(t) is as function of time only. The constraints imposed by pulsating supports due to vertical earthquake excitations can be idealized as harmonic motions of the form y(xi, t) = Ai sin(+
- c(,)
(5)
in which xi is the position of support i along the span, Ai is the amplitude of the ith support settlement, o, is the circular forcing frequency, and a, is the phase angle between the force and the response due to damping. The force at support i for the n th mode of vibration can be written Fh(t) = R, sin w,t,
(
at2 at Fig. 1. General loading and free body diagram.
Mdbc ax v4Kix ax
(6)
949
Vertical support vibration on bridges
where R, is the undetermined amplitude of the equivalent force for support i. The Fourier series expression for F-,(r) is given as F,(l) = zc, A, x,(x),
(7)
where A, is a constant for each mode of vibration and is such that the series will correctly represent F,(t) along the span. Assuming that the concentrated load acts over an infinitely small distance ci as shown in Fig. 2, and multiplying both sides of eqn (7) by mX,(x) in order to use orthogonality relationships we can solve for the A,s
i;.(t)+~q.(t)fp~g,(t)=~,RiX.(x,)sino,(t).
Assuming the solution to eqn (14) to be of the form q.(r) = D,, sin o/r + D,, cos w,f
X,+ c/2 mC, A, -W)&(x) x,-c/2
I
b.
(8)
Y,k
I
x, +
c/2 mFin(r)Xn (x) dx. Xi- e/2
For a beam with multiple intermediate eqivalent load becomes
(9)
(10)
0 = WXxM),
= ZiZnmFi,,(r)Xn(xi)S,,(x), (13)
Fig. 2. Simulation of a concentrated load. CA9 40,4-J
-I1’
/cl
G -m~,~~~~,f
1
. (16)
form of this solution would be
y(x, r) = Y’ sin(o+r - a,)
(17)
where Y’ is the forced amplitude and a, is the phase angle
and
(12)
where q”(r) is a function of time only for the n th mode of vibration. Substituting the value of w(x, t) given in eqn (11) into (1) yields + C, E + m 2
(p;5-wj)sinw#
-,
(11)
The deflection of the continuous beam can also be expressed in terms of the normalized shape function
Hz
(15)
supports the
x(x, r) = CiC,mFin(r)Xn(xi)Xn(x).
Y(-T
x
A more convenient
In the limiting case when c, approaches zero A, = mFh(r)Xn(xi).
W&(x,)
t) = LX(x)
7
For a beam with no energy dissipation at its supports and with a constant rigidity along its span, the shape functions are orthogonal and eqn (8) reduces to A, =
(14)
and substituting this solution and its derivatives back into eqn (15) and then equating sine and cosine terms yields the total particular solution
mF, (r )Xk (x) dx
=
where C,, is used since damping constitutes a nonconservative force system and thus is assumed to be related to the mode of vibration. After some algebraic manipulation using eqns (12), (2), and (3), (13) becomes
~~~
Yp(x, r) = Gf,(x)
x sin&r
- a,)
(20)
J. S. DAVIDSONand C. H. YCMJ
950
which is a system of linear equations that has as many unknowns (&s) as there are interior supports. Because of the complex nature of damping, there are a number of ways to treat the damping coefficient in this problem. But for this development it is assumed that C, remains proportional to pn so that Cn/pn = constant and the ratio to critical damping e, is a constant t.=t
=& n
c”=2p,5. M
(22)
Using this approach to damping and rewriting eqn (20) yields
&RJ”(Xi)
Y,(x, t) = zJ,(x) [
J[
(Pi - w;j2+ (2P”59)*
11
x sin(w,r - a,)
(23)
which is the solution used in the computer program developments. Although the free vibration or homogeneous solution can be determined by the same method used in the forced vibration method, the deflection due to free vibration cannot be taken into consideration in earthquake problems. This is because the initial conditions imposed by these settlements are not exactly known and therefore the amplitudes of the free vibrations cannot be solved. Saibel and D’Appalonia considered multispan beams with pulsating sinusoidal loads acting within the span [5]. In this case, taking the total initial velocity and total initial displacement equal to zero is a reasonable assumption and would hold true for every point along the span, including the support locations. Using these initial conditions, the shape function X,(x) can be factored out and the total solution can be solved. But in the case of multispan beams with forced support settlements induced by vertical earthquake excitations, the motion at the intermediate supports cannot be said to have zero initial forced velocity. Therefore, the total initial velocity cannot be taken as zero for every point along the span. This implies that X,,(x) cannot be factored out as was done in the sinusoidal load case and thus separation of the time and shape functions is not possible for earthquake initial conditions when using Fourier Series derivation. Another limitation of the Fourier Series derivation is that whenever magnitudes of the reactions Ris are equal to zero simultaneously, determining the natural frequencies by setting the determinant of the reaction coefficients to zero is not possible. This is actually the case of the trivial solution to eqn (21) and is not a major drawback because for beams of constant elastic properties, the intermediate reactions will all be zero only in the case of equal span beams. For these cases the frequencies are equal to multiples of
the simple beam frequencies, and can thus be readily determined. 3. NUMERICAL ANALYSIS
Using the formulation presented in Section 2, a computer program has been developed to calculate the deflection, velocity, acceleration, moment, and shear along the span, as well as natural frequencies of the bridge. The program uses eqn (21) to determine the equivalent support settlement reactions induced by the earthquake and eqn (20) and its derivatives to solve for the various functions. Also from this development, design aids can be produced that may be used in industry to design for seismic disturbances. The dynamic amplification factors are computed as the ratios of the maximum dynamic values (deflection, moment, and shear) to the corresponding maximum static values. These charts can be used for bridges that have any combination of E, I, m, and L, but maintain the same relative ratios of intermediate support distances. In order for the design aids to be usable they must be in a nondimensionalized form. The forcing frequency can be expressed as
a’ = &EIymL’)
The dynamic amplification charts are plotted using wi so that they can be used for any bridge that has the same relative support locations but have any combination of E, I, m, and L. The plots were developed by taking the longest intermediate span length to be 1.0 so that all plots could be produced for the same general range of w;. Therefore, in order to use this set of DAF plots, o; given in eqn (24) must be divided by a factor of TL* where TL is the sum of the span ratios given at the top of each DAF plot
w’ = TL2 J(EI/mL4) Damping
In the Fourier series formulation, it was assumed that the damping coefficient varied with the mode of vibration (viscous damping). The ratio to critical damping (5) was defined and used in the computer program developments. But since the Fourier series formulation is based entirely on the vibration modes ,, of the simple beam, the value input for ZETA may not be representative of the actual damping ratio for the multispan beam. The fundamental frequency of the simple beam is always less than the fundamental frequency of the same beam with intermediate supports so in order to get a more representative input value for ZETA the following relation is used
Vertical support vibration on bridges
951
where (27) psimplcis the fundamental frequency of the simple is the fundamental frequency of the b=m, ~~~~~~~~ multispan beam, < is the ratio to critical damping of the multispan beam previously defined (specified at
the top of each DAF plot as ZETA’), and tinputis the ZETA value used int he program input to achieve a more representative response for damping. 4. PARAMETER
-2 i
LezJth 7)
UOsm6)0M
Fig. 3. The maximum dynamic deflection along the span for a support settlement of 0.0805 inches. 200 terns summed.
STUDIES
Example
To illustrate the use of computer programs, consider a two-span concrete bridge of total length 60 ft and width 32 ft. One span of this bridge has a length of 33 ft and the other a length of 27 ft. The bridge has a haunch at the intermediate support. Although the actual bridge will behave in a three-dimensional way, it seems appropriate to model it as a two-dimensional structure using equivalent cross-sectional properties. These properties are [8]: moment of intertia, 92,850 in’.; depth, 17 in.; Young’s modulus, 3,000,OOOpsi; and mass per unit length, 1.467 lb-sec/in2. This bridge is analyzed under the effects of three forcing frequencies and the effects of two damping cases are demonstrated. Although the acceleration rather than the earthquake frequency is usually documented for earthquakes [9], the forcing frequency in this mathematical model can be readily derived by estimating the support settlements under the effect of acceleration. With both the acceleration and support settlement known, the forcing frequency can be determined by CD,=
a
J(> xi
25 33 33
96.6 128.8 128.8
1w
140
uo
400
‘aa
Length(in)
so
040
720
Fig. 4. The maximum dynamic moment along the span for a support settlement of 0.0805 inches. 200 terms summed.
-40
-so
Table 1. Forcing frequencies used in example problems a, ia./se?
a0
--Y)
in which o, is the forcing frequency, a is the acceleration due to the earthquake, Ai is the expected support settlement under the effect of the earthquake acceleration. These parameters are given in Table 1. The dynamic response to these frequencies can be seen in Figs 3-9. From these figures it can be observed that the deflected shape varies drastically with the variation of the forcing frequency, and that for high frequencies the bridge superstructure exhibits more than one bend along its span. With such deflected shapes, the curvature is sharper and thus higher moments and shear.
a, %g
0
A,, in.
o,, rad/sec
.2415 .0805 .0057
20 40 150
lea
7.40
310
400
4ao
MO
MO
720
Length (in)
Fig. 5. The maximum dynamic shear along the span for a support settlement of 0.0805 inches. 200 terms summed.
1::
Length C)
4ao~s4072a
Fig. 6. The maximum dynamic deflection along the span for a support settlement of 0.2415 inches. 200 tenas summed.
J. S. DAVIDSON and C. H. Yoo
952
-10
.
b
.
.
.
m
.
.
.
. . (80
.
.
. 24a
..~I...........I...I~~~ 320 4m
Length
wo
4m
no
Ma
(in)
Fig. 7. The maximum dynamic deflection along the span for a support settlement of 0.0057 inches. 200 terms summed.
-0.7.
6
.
do
lb
.
.
.
. 20
.
.
.
.
.
.
320
Length
.
.
.
.
.
400
.
.
..I.......
sso
4ao
(in)
no
Ma
Fig. 8. The maximum dynamic deflection along the spin for a support settlement of 0.0805 inches, 200 terms summed.
Fig.
10. Dynamic
b
do
#.
?@a
I..
240
8..
am
8.
400
Length(in)
..
I
4ao
”
I
so
“‘I
”
MO
‘110
Fig. 9. The maximum dynamic deflection along the span for a support settlement of 0.0805 inches. 200 terms summed.
For a forcing frequency of 40 rad/sec, the maximum deflection occurs at 540 in. (45 ft) from the left abutment; the maximum moment at 568.8in. (47.4 ft); and the maximum shear occurs at 410.4 in. (34.2 ft). The fundamental natural frequency of this bridge is computed as 31.43 rad/sec. Now using this same example to illustrate the use of the dynamic amplification charts, the maximum static deflection is 0.081045 in., the maximum static moment is 524300 lb-in., and the maximum static shear is 1618.4. The value of wi for an earthquake frequency of 40 rad/sec can be obtained from eqn (25) 40
fBj= 1.8182’
3,000,000 x 92,850 1.467 (720)4 >
factors
for deflection
Entering the deflection chart Fig. 10 with ZETA = 0 (no damping), the dynamic amplification factor is 2.15 and thus the maximum dynamic deflection expected is 2.15 x 0.081045 = 1.742 in. Other DAF for deflections are given in Figs 11 and 12 for nonzero damping cases. The exact value as produced by the Fourier series program is 1.738 in. The DAF for the moment can be obtained from Fig. 13 as 7.0 and hence the maximum dynamic moment is 7.0 x 524,300= 3,670,100 lb-in. The exact value is 3,649,300 lb-in. The dynamic amplification factor for shear is 36.0 as can be read from Fig. 14 and the maximum dynamic shear is therefore 36 x 1618.2 = 58255 Ibs. The exact value is 55457 Ibs. 5.
-0.7.
amplification functions.
DISCUSSION
OF RESULTS
From the example given in Section 4, and from other examples considered, a good correlation was observed in comparing the results of the DYNAMIC computer program to the results achieved by using the dynamic amplification method. Also, for cases in which damping forces are neglected, it has been shown that a very good correlation exists between the results obtained from using a three-moment equation development and the results obtained from using the Fourier series [lo]. It was observed, however, that more terms have to be summed as the number of bridge spans is increased. For example, it was observed that the summation of the first 50 terms gave reliable results for the frequency computations
Q
= 14.39. Fig.
11. Dynamic
amplification factors functions.
for deflection
953
Vertical support vibration on bridges
Fig.
12. Dynamic
amplification functions.
factors
for deflection
moment amplification, and a substantial increase in the shear amplification values. This is primarily because the shear is related to the rate of curvature which is increased with the higher forcing frequencies. A tinite element analysis was performed on the example presented in Section 4 using ADINA. A modal analysis was used to verify the natural frequencies produced by the Fourier series program and direct integration was used to produce the maximum absolute values of displacement, moment, and shear along the span for the three forcing frequencies used in Section 4 (see Figs 15-19). In general, a good correlation was noted between the deflected shape of
(........................... ........
-0.4
0
a0
180
ma
310
Length
Fig.
13. Dynamic
amplification functions.
factors
for
moment
400
am
I . . . . . . . . . . . . . . . . . . . . . . . . no Ia0 a40 ?no 400 4aa Lmgth (in)
0
as well as for the deflection, moment, and shear functions for two span bridges, whereas 100 and 200 terms are needed to achieve the same accuracy for three- and four-span bridges, respectively. As expected, the convergence of the deflection function is the most rapid due to the existence of the n4 term in the denominator of the deflection expression. The convergence of the moment function is faster than that of the shear function because of the nz term in the denominator of the moment expression as opposed to the n term in the shear expression. In the DAF plots, damping causes a sharp peak at resonance locations with some peaks higher than others. Neglecting the resonance effects, the DAF plots show little increase in deflection amplification with increase in forcing frequency, some increase in
w
no
Fig. 15. The maximum absolute dynamic deflection along span for a support settlement bf 0.0805 inches. Using ADINA.
-10.0
Fig. 14. Dynamic amplification factors for shear functions.
MO
(ill)
..,...,...i MO
640
no
Fig. 16. The maximum absolute dynamic moment along the span for a support settlement of 0.0805 inches. Using ADINA.
--3oo --4oo
1! . . . . . . . . . . . . . . . . . . . . . . . .._... 0
a0
1e4
240
w Length
400
4m
am
_.,_ Ma
no
(In)
Fig. 17. The maximum absolute dynamic shear along the span for a support settlement of 0.0805 inches. Using ADINA.
J. S. DAVIDSON and C. H. Yoo
954
Fig. 18. ine maximum absolute dynamic deflection along the span for a support settlement of 0.2415 inches. Using ADINA.
--P),...,...,...,...,...,...,...,...,... 0
aa
100
240
320
Length
400 (in)
4m
SW
e4a
no
Fig. 19. The maximum absolute dynamic deflection along the span for a support settlement of 0.0057 inches. Using ADINA. the structures and magnitudes of the displacement, moment, and shear. The model used in ADINA produced higher values of deflection, moment, and shear, but within the same order of magnitude. 6. SUMMARY AND CONCLUDING REMARKS The response of highway bridge type structures to dynamic differential support settlement was investigated. The bridge was modeled as a two-dimensional continuous beam with pinned ends and intermediate supports. The governing differential equation of motion (including damping) was derived for pulsating support settlements induced by the vertical acceleration of an earthquake. The deflection, moment, shear, velocity, and acceleration expressions were determined in terms of a Fourier series expression and a computer program was developed to calculate these values. The results of this development were compared to the results of a finite element analysis (ADINA) and a good correlation was noted between deflected shapes and the absolute maximum displacement, moment, and shear values along the span. The parameters in the Fourier series expression were nondimensionalized and plots were created for dynamic amplification factors versus forcing frequency. A two span bridge example was used to illustrate the use of these plots as design aids and a good
correlation was noted between the use of the design aids and the results produced by the Fourier series program. Further investigation should be done on the use of damping in this development. Presently only viscous damping is considered and it is assumed to be proportional to the simple beam frequencies. The peaks in the dynamic amplification plots, indicating dynamic resonances, may not be feasible since considerably dynamic oscillation (duration of vibration) is normally required to develop a full resonance effect. Also, actual earthquake motion is by no means a simple harmonic motion. A study correlating the validity of using simple harmonic motion to represent the structural behavior of the system subjected to an earthquake may be of value. Past performances of bridges during earthquakes such as the San Fernando earthquake [l l-131 suggest that a majority of the bridges which collapsed were subjected to unusually high shear. This investigation verified those observations. As can be seen in the charts, the dynamic amplification factors for shear are quite high, especially under higher forcing frequencies. Until the adoption of more rigid design criteria, a few design hints based on some general observations in this study might be appropriate. Since the dynamic amplification factors for shear were the highest, and probably controlling for all ranges of forcing frequencies, strict attention should be given to the details of the shear resisting members. Also, since the natural frequency is inversely proportional to the second power of the total length of the bridge, bridges with shorter span lengths should be considered in areas of high seismic activity. This increases the natural frequency of the bridge and decreases the dynamic amplification factors. Acknowledgements-Support of this work was partially provided by the National Science Foundation through Grant No. PFR-7822845. The support of Dr John B. Scalzi of NSF is gratefully acknowledged. REFERENCES
1. USCGS, United States Earthquakes of 1956. U.S. Government Printing Office, Washington, DC (1958). 2. M. Chen and J. Penzien, An investigation of the effectiveness of existing bridge design methodology in providing adequate structural resistance to seismic disturbance. Phase III, Report No. FHWA-RD-75-10, University of California, Berkeley (1974). 3. W. Tsen and J. Penxien, An investigation of the effectiveness of existing bridge design methodology in providing adequate structural resistance to seismic disturbance. Phase II, Report No. FHWA-RD-74-3, University of California, Berkeley (1974). 4. G. L. Rogers, Dynamics of Framed Structures. John Wiley, New York (1959). 5. E. Saibel and E. D’Appolonia, Forced vibrations of continuous beams. Trans. ASCE 117,1075-1090 (1952). 6. M. M. Biggs, Introduction to Structural Dynamics. McGraw-Hill, New York (1964). 7. W. Nowacki, Dynamics of Elastic Structures. John Wiley, New York (1973).
Vertical support vibration on bridges 8. R. J. Banerian, An analysis of the behavior of short two span and three span highway bridges subject to pulsating support settlements. Thesis presented to Marquette University, at Milwaukee, Wisconsin, in partial fulfilment of the requirements for the degree of Master of Science (1977). 9. R. L. Wiegel, Earfhquuke Engineering. Prentice-Hall, Englewood Cliffs, NJ (1970). 10. S. V. Acra, Effect of vertical vibration of the supports of a highway bridge on a superstructure. Thesis presented to Marquette University, at Mil-
955
waukee, Wisconsin, in partial fulfilment of the requirements for the degree of Master of Science (1980). 11. A. L. Elliott, Hindsight and foresight on the performante of prestressed concrete bridges in the San Fernando earthquake. PCI JI 17, (1972). 12. H. S. Lew et al., Engineering aspects of the 1971 San Fernando earthquake. BSS40, NBS, Washington, DC (1971). 13. W. F. Pond, Performance on bridges during San Fernando earthquake. PCI JI 17, No. 4 (1972).