Performance and New Concepts for Earthquake Protective Systems
(Tasks D1-1, D1-2, and D2-1)
George C. Lee, Multidisciplinary Center for Earthquake Engineering Research
Zach Liang, State University of New York at Buffalo
Task D1-1 is a review of the historical performance of existing EPS - bridges in recent earthquakes. The earthquakes included Kobe, Northridge, Turkey, and Taiwan. Information on one additional earthquake (Iceland, 2000) is being sought.
Task D1-2 is on theory and design. We have carried out some in-depth studies with respect to the dynamic responses of bridges cross effects so that EPS can be properly designed. The theoretical results are equally applicable to buildings and other structures.
Task D2-1 is to develop and evaluate prototype intelligent passive systems. A first attempt is to conceive a new type of sloping surface sliding isolation system. A prototype is made to demonstrate its efficiency and other advantages. A U.S. patent is being filed by the Technology Transfer Office of the University at Buffalo.
Earthquake Performance of Bridge Bearings in Recent Earthquakes
Since the 1994 Northridge earthquake, several major earthquakes had occurred. They include the 1995 Kobe earthquake, the 1999 Turkey and Taiwan earthquakes and the 2000 Iceland earthquake. All of these earthquakes involve heavy damages (or collapses) to bridges. We have reviewed all possible information available in the open literature on performances of bridge bearings, both regular or isolation bearings. Information of the performance of several bridges during the Year 2000 Iceland earthquake will be obtained during the month of October 2000. It is anticipated that a technical report summarizing our study will be completed before the end of the Year 2000 for review. This report will include performances of the bridge bearings since the 1994 Northridge earthquake. Approximately forty references will be included. This list of references was presented in the Jan. 1 - March 31, 2000 Quarterly Report. It did not include information of the Turkey, Taiwan and Iceland earthquakes.
Theoretical Study on Cross Effect
Cross effect is the relationship between loading and responses in two mutually perpendicular directions of a structure. The first year research is focused on linear relationships. There are two important reasons that have motivated us to pursue this line of research study.
First, the existence of cross effect, needs to be demonstrated because cross effect can significantly magnify structural responses. During this year's study, we have shown that, both theoretically and experimentally, cross effect did exist at virtually any engineering structures and that the effect could be significant.
Second, cross effect needs to be quantified. Although for many structures cross effect is small, so that ignoring this effect would not introduce much design error. However, we have shown that this error can be very large. In particular, we must know the magnitude of cross effect in order to properly design base isolation and other EPS for bridges.
The study on cross effect is aimed for engineer applications, especially for highway bridges. In the following, some important conclusions for this first year study will be given
It is known that a single structural member always has a pair of principal axes along which a load will not cause perpendicular responses. However, a structure such as a bridge or a high-rise building frame may not possess a pair of principal axes throughout its length. In such a case, along any direction a load is applied at a certain location, in the perpendicular direction at another location, there may exist a response. It is referred to as the cross effect between the two locations in perpendicular directions. More specifically, we may call it the cross effect of flexibility. Cross effect of stiffness is the other type of cross effect. In that case, no matter which direction is chosen to measure a deformation at a certain location, there is always a force involved at another location in the perpendicular direction. In these circumstances, there will be certain amount of energy transferred between these two perpendicular directions.
A structure has no static cross effect does not necessarily mean that it has no dynamic cross effect. Static cross effect is caused by non-existing of principal axes of stiffness, which is the stiffness cross effect. In the following, if not specified, both stiffness and flexibility cross effects are discussed at the same time.
When a structure is applied with a static load and cross effect occurs, it is referred to as the static cross effect. When the load is dynamic, if cross effect exists, then it is referred to as the dynamic cross effect. Usually, the static cross effect can be quite small. However, the corresponding dynamic cross effect of the same structure may be magnified with a factory of 30% or even more because the dynamic responses can be a process of energy accumulation.
Dynamically, even both stiffness and damping coefficient matrices have their own principal axes, when these two set of principal axes are not the same, cross effect exists.
Most structures with four or more degree-of-freedoms are likely to have no principal axes. If the load-response problem of a structure cannot be constrained within one plane, then the structure is likely to have no principal axes. For example, a bridge is supported at two locations. Each of them has principal axes locally but the principal directions are different. In such a case, this bridge will not have global principal axes.
The theoretical development can be explained by using a two-dimensional loading-deformation problems for reason of simplicity. They can be extended to three-dimensional problems.
A global stiffness matrix, obtained through a finite element model or other methods, can be rearranged to have four partitioned sub-matrices, two diagonal and two off-diagonal matrices, and denoted by
, Upper diagonal, standing for the case of load applied in X direction and responses measured also along X direction;
, Lower diagonal, standing for the case of load applied in Y direction and responses measured also along Y direction;
, lower off-diagonal, standing for the case of load applied in Y direction and responses measured along X direction; and
, upper off diagonal, standing for the case of load applied in X direction and responses measured also along Y direction.
In the above the subscriptions F(.) denotes the force is allied in the direction of (.) and D(.) stands for the deformation or displacement is measured along direction (.). And in the following, we use X and Y to indicate the pair of perpendicular directions.
Similarly, we can have partitioned flexibility matrices denoted by respectively.
It is noted that, according to the Maxwell Law, we have
where the superscript T stands for matrix transpose. The above equation means that the two diagonal matrices are symmetric. Generally, we have
but we may not always have
Similarly, we may have the partitioned flexibility sub-matrices and
Similarly, we usually may not have
The symmetry described by equations (1) have special meanings. For example, (1b) implies that between any location i and j, we have the following relation: Suppose a unit load is applied along direction Y at location i and a measurement of deformation is taken along direction X at location j. Also suppose a unit load is applied along direction X at location i and a measurement of deformation is taken along direction Y at location j. These two deformations are identical. It can be shown that symmetry exists for the case of trusses, etc.
To determine the principle directions of a structure is equivalent to find an angle theta to have an orthonormal matrix Q
and to construct a new stiffness matrix Knew such that
The above equation means that the load is applied and deformation is measured with a rotation of angle ? ??from the original directions. The same is true for the flexibility matrices. To determine if a structure has principal axes is to find whether the angle ? exists. If ? does exist, then "rotating" the direction of load by that angle we can find the principal directions. And in such a case, the off-diagonal matrix
and so on. It is also true for flexibility matrix. Equation (2) indicates that in the principal directions, the structure is decoupleable between the two perpendicular directions.
It is noted that, if the stiffness matrix has principal directions, the flexibility matrix has the same directions.
In the principal directions, we also have
trace () = max.
trace () = min. (3)
The symbol max. and min. stand for the value of the trace reaches the maximum and minimum points. In such case, axis X is referred to as the major axis and axis Y is referred to as the minor axis. Comparing to a plane problem with two DOF, it is known that in the major axis, we have the strongest stiffness and in the minor axis we have the weakest, about which the cross terms vanish. In general cases, when a structure is "rotated" to its principal directions, the off-diagonal matrix will also become null. The iith diagonal entry of the diagonal sub-matrix indicates the loading and response at location i are along one of the principal axis. The trace of the diagonal matrix is the summation of the diagonal entries. In the principal direction, one of the summations reaches the maximum valve whereas another one becomes the least. This phenomenon defines the principal axes clearly. However, it is worthy to note that an individual entry in the principal direction does not necessarily reach its maximum/minimum value. Only the summations do.
Generally, a stiffness or flexibility matrix with rank 4 or more cannot be decoupleable in two perpendicular directions. That is, equation (2) and the like do not hold. However, any stiffness or flexibility matrix can be rotated in a certain pair of orthogonal directions along them equation (3) hold. In addition, we have
trace ( ) = 0 (4)
From equations (2) and (4) we can define the pseudo-principal axes for non-decoupleable stiffness (flexibility). That is, when the pseudo principal directions are reached, the summation of diagonal sub-matrix reaches its extreme value and the summation of the off-diagonal sub-matrix becomes zero. In this case, we also refer X axis as the major axis and Y to be the minor axis.
Generally speaking, if equation (1a) or (1b) does not hold, then the structure will have no true principal axes. Thus, these equations can be used as criteria as necessary conditions to determine if a structure is decoupleable. We define the off-diagonal sub-matrix as the cross-matrix. If condition (2) is reached, the structure is said to be orthogonally symmetric.
When we "rotate" the stiffness or flexibility matrix, generally all the entries of the matrix will be changed. However, there exist certain invariables.
First, the trace of such matrix is invariable. Secondly, different ijth and jith entries of the cross-matrix is invariable. This may be explained by using the flexibility matrix. This leads to the establishment of the limitations of Maxwell's Law when it is used in dynamic response analysis. Details will be presented in a technical report co-authored by Z. Liang and G. C. Lee.
Sloping Surface Isolation Bearing
We began Task D2-1 in year one to review and evaluate available information in the open literature on smart materials and control technologies including all MCEER efforts. The purpose of this review is to determine possible available technologies that can be adopted for bridge seismic response modifications. A small group of researchers had brain stormed about the desirable features of bridge isolation systems. They came up with the idea of proceed along the direction of developing a mechanical system with sloping surfaces. Since then they moved swiftly to design and fabricated a demonstration model by using steel rollers and called it Sloping Roller Isolation System (SRIS). They made an invention disclosure to the officials of the University at Buffalo and moved on with laboratory evaluations. Test results have been reported in the 4/1/00 - 6/30/00 Quarterly Report. The major motivations for developing the Sloping Surface Isolation System (SSIS) by using steel are:
- Reduction of large superstructure acceleration
- Tolerance of large base travel
- Low fabrication and maintenance cost.
Prototype Bearings and Experimental Test
We have conducted tests on three different setups. The first one is a low friction setup without added mass. The second one is a low friction setup with a heavy added mass. The third one is high friction setup with the added mass.
The purpose of the first test is to study the performance of the sloping bearing. It is also designed to verify the theory that the entire system should not have visible natural frequencies.
The second test is to compare the acceleration reduction without and with heavy mass, which is an effort to verify the theory. It is also designed to study the stability of SRIS with light damping.
The third test is to study the effectiveness of SRIS in terms of acceleration reduction with given allowance of relative displacement, This setup is to simulate a real working condition of SRIS.
The following are certain parameters and/or quantities used in the above test setups and major characters of the prototype bearings:
Angle of slope 4.5o Length of slope 12 in Weight of original steel frame 3000 lb. Weight of added mass 25000 lb. Designed friction coefficient without damping added 0.005 in for rolling 0.1 for sliding Spring constant of each leaf springs 13 Kips/in Maximum deformation of leaf spring 0.7 in Spring constant of each urethane buffer 5 Kips/in Maximum deformation of urethane buffer 0.2 in Young's modulus of metal parts of the slop and rollers 30,000 Kips/in2 Yielding stress of the metal parts 80 kips/in2 Diameters of rollers 1.25 in Length of roller 2.5 in Number of rollers in each bearing 16 Number of total bearings 4 Maximum allowable travel of bearing 12 in Dimension of bearing (L x W x H) 18 in x 18in x 11 in Total test for sinusoidal input 40 Total test for earthquake input 340
Figure 1 shows the conceptual assembly of the SRIS bearing. The bearing consists of three major parts, and two perpendicular layers of rollers, so that the upper and bottom parts can have relative movement in any horizontal directions.
In each layer, there are two sets of rollers traveling along tracks as shown in X shaped configurations. The tracks break the single conventional pendulum surface into two tracks so that when the bearing starts to move, the roller moves from one side of the bearing reaching the another side. In conventional design, the movement starts at the center of the bearing. Thus, the proposed design can save at least one third of the travel distance, which in turn tolerates considerably larger bearing displacements.
The roller sets and the corresponding tracks can be replaced by sliding surfaces, which can bear large vertical load. However, the roller set has a significant advantage because the rolling friction is much smaller than sliding friction. The angle of the track of the roller system can therefore be as small as 1.50 to 20. The angle of the track directly affect the level of acceleration of superstructures. The smaller the angle, the better results will be achieved.
Comparing with standard pendulum bearings, the angle of the track of SRIS is constant whereas a pendulum bearing will virtually have no slope near the center. A certain slope is required to restore the center position to avoid permanent bearing displacement. Apparently, the X shaped track has advantage for self-centering features.
The experimental test was conducted by using the earthquake shaking table at U.B. In the test setup, a mass is mounted above four SRIS bearings. The SRIS bearing set must be so arranged. They cannot be used separately. The bearings are installed at the 450 to the direction of table motions to test the function of bearing movement along arbitrary directions. The test has two phases, with and without a heavy mass.
The input excitations include all available earthquake records at the MCEER laboratory. For most records, we conducted tests with the full scale of g's levels or higher. However, certain records with long period will exceed the limitation of allowed displacement of the shaking table. In order to maintain the amplitude, the time scales were reduced.
In the following, we briefly present the major test results. Detailed data and plots will be available in the final report of the SRIS test.
The first test is for sinusoidal excitation with different driving frequencies. With the range of 1 Hz up to 4 Hz, the response of the superstructure is virtually a flat line without obvious peaks. This proves our theory on the nonlinear behavior of SRIS that makes the responses vs. driving frequency a straight line relationship. This is a very important feature of the SRIS bearing, which implies no notable peak values of resonance. Therefore, unlike most existing isolation bearings which can magnify the superstructure's accelerations instead of reduce them, SRIS will guarantee the acceleration reduction even in rather low frequency ranges. Therefore, the proposed bearing can be suitable for long period earthquakes.
The relative displacement between the super structure and the shaking table is also examined thoroughly three plots; the input absolute displacement, the relative displacement and the ratio of the relative displacement over the input are generated and studied with respect to the straight line relationship mentioned earlier.
We also obtained the results under the testing condition of both horizontal and vertical excitations. SRIS performs well in term of acceleration reduction on any existing earthquake records. And the corresponding relative displacements are also reduced. These results are different from the sweep-sine test.
A sinusoidal test series with variable amplitude of input but fixed driving frequency is also conducted . No matter how the amplitude increased or decreased, the output of the acceleration of the super structure remains virtually at the same level. This proves the theory of non-linearity of SRIS, which can be beneficial in practical applications.
We also examined the normalized force with respect to the weight of super structure vs., the relative displacement. Such relationships are observed when both sinusoidal and different earthquake inputs are used. This test is intended for the study of actual energy dissipations. When the input level changes, the energy dissipation remains unchanged. The energy dissipation can be adjusted by additional dampers, either passitive or semi-active. Note that, energy dissipation achieved by using a comparatively larger friction force is not the purpose in SRIS design. To achieve good results of acceleration reduction, we actually do not have to use large damping. Adjusting the energy dissipation can vary the rate of acceleration reduction and the level of displacement. In a certain circumstance, we need to sacrifice a certain amount of acceleration reduction in order to achieve smaller displacement. In SRIS we thus have the measure to do so.
All the tests with different excitation records have shown that the level of acceleration of the superstructure are regulated below 0.15g. The input level, however, are from 0.2 g up to 0.82 g. This promises a significant reduction. In addition, it exhibits a beneficial non-linearity for structure designers. Finally, large relative displacement can be tolerated, which is important in design.
In order to ensure accuracy, we also compared tests with and without reducing the time scale. The reason to use such records is entirely because of the limitation of the shaking table, which can only allow a certain displacement. Therefore, when the driving period are long, we are forced to reduce the time scale to decrease the amplitude of the ground displacement.
For this task, a US patent is being applied and commercialization partners are being sought by the University's Technology Transfer Office. A technical report is under preparation to describe the principles and experimental results obtained to date, by Dr.'s Lee, Liang and Niu, who are the co-inventors of SSIS.