was conducted by George W. Housner, who investigated the mechanics and ..... Gajan, S., B. L. Kutter, J. D. Phalen, T. C. Hutchinson and G. R. Martin (2005).
Insitu free vibration tests on a single degree of freedom structure with foundation uplift T. B. Algie, R. S. Henry & Q. T. Ma Department of Civil & Environmental Engineering, University of Auckland, New Zealand 2007 NZSEE Conference
ABSTRACT: A series of free vibration tests were conducted on a free standing single degree of freedom structure with a range of elastic natural frequencies. The tests were conducted over a rigid concrete strong floor and on Auckland soil. A typical response was obtained when the structure was displaced from its original alignment, held still, then released and allowed to rock. The response as tested over the concrete strong floor fitted Housner’s simple rocking model, however it required an empirical manipulation of the damping factors. Interestingly, the experimental result showed that the amount of radiation damping, measured as the apparent co-efficient of restitution, is relatively constant irrespective of the structure’s geometry, which contradicts the traditional rocking models. The results showed that when the structure rocked on insitu ground, the dynamic behaviour was greatly modified. The time history response demonstrated a change of principle damping scheme from radiation to viscous. The response over the continuous medium failed to correlate with Housner’s model due to the increased viscous damping, which is attributed to the soil absorbing energy through deformation. An elongation in the rocking period for a given amplitude was observed. 1 INTRODUCTION One of the first comprehensive studies of the mechanics of rocking as applied to structural dynamics was conducted by George W. Housner, who investigated the mechanics and seismic response of a rigid rocking block (Housner 1963). By studying the rigid rocking block, Housner deduced that tall slender structures when allowed to uplift from their foundations, are more stable than anticipated from overturning and provided an effective isolation mechanism from ground motions. But despite the rocking problem’s simple appearance, researchers have shown repeatedly that it is also a highly nonlinear and sensitive problem with chaotic traits. (Bruhn and Koch 1991, Yim and Lin 1991).The rocking block was found to have five modes of response: rest, rocking, sliding, rocking-sliding and free-flight (Shenton and Jones 1991). One of the more recent applications of rocking for seismic protection is found in the Precast Seismic Structural System (PRESSS) research program, as part of an effort to improve the seismic performance of precast/prestressed concrete buildings in the 1990s (Priestley et al. 1999). In this program, extensive static and pseudodyanmic testing were conducted and the jointed rocking wall system was demonstrated as a highly competent lateral load resisting system, resulting in a residual drift of only 0.06% following a peak drift of 1.8% under a 1.5 times design level earthquake. Although there have been numerous efforts to model the dynamic response of rocking walls, shake table testing of post-tensioned concrete masonry (PCM) walls has revealed the shortcomings in these current analytical efforts (Wight et al. 2004). Studies carried out at the University of Auckland highlighted the inaccuracy of modelling the damping factor of these systems using standard finite element procedures (Ma et al. 2006). This current study aims to provide additional high quality dynamic experimental data to further the investigation of the effects of the interface material on a rocking structure’s dynamic response (ElGawady et al. 2006a; ElGawady et al. 2006b). Recent studies into shallow foundations at the University of California, Davis, found that during rocking subsoil can be forced into a nonlinear range. This soil structure interaction further complicates analytical efforts to model these structures (Gajan et al. 2005).
Paper Number 25
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In this current study, a series of free vibration tests were carried out on a single degree of freedom (sdof) column with various elastic natural frequencies. The tests were conducted under three restraint conditions on the concrete pad footing: 1. Footing rigidly fixed, no rocking was allowed 2. Footing allowed to rock on a rigid concrete strong floor 3. Footing allowed to rock on insitu ground These restraint conditions were designed to allow preliminary investigations into the behaviour of structures that are allowed to rock on their foundations as opposed to a rigid interface medium. Additionally, it was intended to investigate the suitability of simple rocking models (SRM), similar to that proposed by Housner to predict the response of such structures. 2 BACKGROUND The response of the single degree of freedom rocking column was compared with the simple rocking model (SRM) proposed by Housner (Housner 1963). A differential equation of motion for free rocking response was developed by equating overturning and restoring moments of a rigid rocking block as shown in Fig. 1. The equation of motion as shown in Eq. 1 below can also be applied to a rocking sdof column with suitable substitutions.
Io where
d 2θ = −WR sin(α − θ ) dt 2
(1)
Io = moment of inertia about origin O (kgm2); θ = angular displacement (rad); W = weight of the block (N); R = distance from origin O to centre of gravity (m); α = slenderness angle (rad). 2B
2H
θ W
R
α
O’
O
Figure 1. Housner’s Rocking Block
Then by applying conservation of momentum, Housner predicted the coefficient of restitution; Eq. 2, which allows for the reduction in velocity caused at impacts.
⎛ ⋅ ⎜θ r = ⎜ ⋅2 ⎜θ ⎝ 1
2
⎞ ⎤ ⎡ mR 2 ⎟ ( ) 1 1 cos( 2 α ) = − − ⎥ ⎢ ⎟⎟ Io ⎦ ⎣ ⎠
(2)
⋅
where θ = angular velocity (rad/s); and m = mass of block (kg).
2
Finally, by assuming a tall slender structure and under small angular displacements, Housner developed a prediction for the period of a rocking cycle given in Eq. 3.
T=
⎞ ⎛ 4 1 ⎟⎟ cosh −1 ⎜⎜ p ⎝1 − θo α ⎠
where T = period of rocking cycle; p =
(3)
WR / I ; and θo = initial angular displacement.
3 EXPERIMENTAL SETUP Three series of free vibration experiments were conducted on a 1.9m high, steel 100UC14.8 column. A 300x300x100mm concrete footing was cast and the column was secured to the footing using 2 M20 bolts via a steel baseplate. The concrete footing was trimmed by steel angles which prevented any damage. Five concrete mass blocks were cast to simulate different seismic masses and were placed at the top of the column. These blocks provided a mass of up to 87kg. The arrangement is shown in Fig 2.
Figure 2. The column test rig used to conduct free vibration/rocking experiments and the three restraint conditions; rigid footing, rocking on concrete and rocking on soil
The first series of free vibration tests was conducted with the column footing rigidly fixed to the concrete strong floor. In the second series of tests, the footing was released from the concrete strong floor and allowed to uplift, As expected, a rocking response ensued. Two steel plates were installed on both sides of the footing to prevent the footing from sliding. In this series of tests, the column was manually displaced by 120mm and released. Finally, the test rig was relocated outside the laboratory and the test repeated with the footing situated on actual soil. In all three test scenarios, the tests were repeated with five concrete mass blocks (87kg) and three concrete mass blocks (52kg). This allowed the effect of modifying seismic mass and hence elastic natural frequency under rocking motion to be investigated. To record the response on the column, two Linear Variable Differential Transformer’s (LVDT’s) were installed at the top and bottom of the seismic mass to measure horizontal displacements, which allowed angular rotation, θ, to be calculated. An accelerometer was fitted to the top of the rig and the
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accelerations in the rocking plane of the column were measured. Lastly, two strain portal gauges were mounted on the rocking edges of the footing to record the uplift and rotation. 4 EXPERIMENTAL RESULTS 4.1 Free Vibration Tests with Footing Rigidly Fixed (Elastic Column Tests) A typical displacement time history measured below the mass blocks under free vibration is shown in Fig. 3a. A natural period of vibration of 0.274s was recorded with five seismic mass blocks (87kg). Additionally, the viscous damping was calculated to be 1.4%. These results compared well with a 10 element, finite element model created using SAP2000, which predicted a first mode natural period of 0.249s. The small discrepancy can be attributed to the flexibility of the baseplate connection. The SAP model assumed a rigid restraint at the column base whereas the experimental data from the portal gauges attached to the footing, showed that a rotation of up to 0.2 degrees occurred at this connection. Displacment at Top
Comparison of Rocking on Strongfloor and Soil
0.06
30
Strongfloor Soil
Elastic test (5 weights) 0.05 0.04
Angular Displacement (rad)
Displacement (mm)
20
10
0
-10
0.03
0.02 0.01 0
-0.01 -0.02
-20
-0.03 -30
0
1
2
3
4
5
6
7
8
9
-0.04
10
time (s)
0
1
2
3
4
5
time (s)
6
7
8
9
10
Figure 3. 3a) A typical displacement time history response for an elastic column test and; 3b) A typical displacement time history response for free rocking tests on concrete and soil
4.2 Comparisons of Rocking Interface Material The time history responses of the column rocking on concrete (an ideal rigid surface) and soil (continuous medium) were found to be very different. This is clearly illustrated in fig. 3b, which shows the rotational time history of two tests with the same initial displacements but dissimilar interface materials. The column situated over a continuous medium (soil) showed a significant increased rate of decay and hence increased energy dissipation. Simplistic analysis equated the results to an average equivalent viscous damping ratio of 11% for the tests on concrete and 22% for the soil. Additionally, the differences in the response curve during the first quarter cycle in Fig. 3b showed that the soil interface in fact has an additional viscous damping source that acts over the whole cycle and not just at the times of impact as is the case with the concrete interface. This phenomenon suggests the form of principle damping has shifted from radiation to viscous damping and can be attributed to the deformations in the continuous medium as the footing rotates. Initial observations suggest that there is a slight elongation in the rocking period from the concrete to soil response. This supports previous research that suggest soil-structure interactions can attribute to unaccounted for seismic resistance through an increase in damping and a minor elongation of the natural period. Note however that too long a period can have a negative effect on a structure. It can also be seen from the footing edge uplift data as shown in Fig. 4, that the critical rocking edge in contact with the ground dips below the original ground level during each cycle. At the end of motion, permanent deformations exist at the rocking edges. This shows that the soil was deformed into the nonlinear range during rocking, obviously an additional source of damping as postulated in previous time history analyses.
4
16 Uplift North Edge Uplift South Edge
14 12 10
Uplift (mm)
8 6 4 2 0 -2 -4
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time (s)
Figure 4. Vertical displacements of critical edges of the footing when rocked on soil
Finally, horizontal acceleration data from the top of the column, as plotted in Fig. 5, show the soil interface was able to significantly decrease the large acceleration pulses resulting from impacts, while at the same time dissipating more energy than the rigid interface (concrete), as demonstrated by the increased overall damping in the time history response. Horizontal Acceleration
8
Horizontal Acceleration
1.5
Rocking on Soil (5 weights)
Rocking on Strongfloor (5 weights) 6
1
Acceleration (ms2)
Acceleration (ms2)
4
2
0
-2
0.5
0
-0.5
-4
-1 -6
-8
0
1
2
3
4
5
6
7
8
9
-1.5
10
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
time (s)
time (s)
Figure 5. Horizontal component accelerations for the column rocking on (a) concrete and (b) soil
4.3 Effect of Variable Seismic Mass As noted previously, the free vibration tests were repeated with two seismic mass conditions, five weights (87kg) and three weights (52kg). The typical time history response for both setups, freely rocking from the same initial displacement on the concrete floor is shown in Fig. 6. The peak amplitudes and half periods were calculated for both setups and were plotted for comparison in Fig. 7. This comparison showed that there is a slight shortening in the rocking period for a given initial displacement, as the seismic mass is reduced. This was to be expected and it is predicted by Housner’s SRM, as the value p in Eq.3 is reduced due to the lowering of the centre of gravity and the reduction in I.
5
Angular Displacement vs Time
0.06
5 Weights 3 Weights
0.05
Angular Displacement (rad)
0.04 0.03
0.02 0.01 0
-0.01 -0.02 -0.03 -0.04
0
1
2
3
4
5
Time (s)
6
7
8
9
10
Figure 6. Angular rotation time histories of column with three and five weights rocking on concrete
However, the similarity in the decay in peak amplitudes in both tests was unexpected. The results suggest that the energy dissipation at each impact for both setups was similar. This suggests the coefficient of restitution (r) is insensitive to the change in rig properties, which disagrees with Eq. 2 from the SRM. Half Period Comparison
Peak Amplitude Comparison 0.06
5 Weights 3 Weights
0.05
1
0.04
0.8
Tn/2 (s)
Peak Theta (rad)
5 Weights 3 Weights
0.03
0.6
0.02
0.4
0.01
0.2
0
0
1
2
3
4
5
6
7
8
9
0
10
1
2
3
4
5
6
7
8
9
10
Number of Impacts
Number of Impacts
Figure 7. Amplitude and period comparison of variable seismic mass rocking on concrete
4.4 SRM Applied to Rocking on Concrete Strong Floor A comparison of the simulated and actual displacement time histories for tests when the column was freely rocking on the concrete floor is shown in Fig. 8. The simulation was first conducted using the apparent coefficient of restitution, rH = 0.973, as predicted by Housner in Eq. 2. Fig. 8 shows that this approach gave a very poor estimation. Subsequently, the r value was adjusted to achieve a ‘best possible fit’. It was found that when the r value was empirically set to 0.757 the simulation best matched the experimental data. This required r value is 22% below that predicted by Housner (rH) implying that more energy was dissipated at each impact above the assumption of a fully plastic impact.
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Angular Displacement vs Time
0.06
Housner Style ODE Solution r=0.757 Housner Style ODE Solution r=rH Test Data
Angular Displacement (rad)
0.04
0.02
0
-0.02
-0.04
-0.06
0
1
2
3
4
5
Time (s)
6
7
8
9
10
Figure 8. Time history response of SRM simulations compared with tests data on concrete strong floor
Additionally, it can be observed from Fig. 8 that during the first quarter cycle the simulated responses were identical for all r values. This was because no impact has yet occurred and thus during this time the theoretical response is independent of the coefficient of restitution (r). The SRM simulations were repeated for the tests with three seismic mass blocks (52kg). It was found that for these tests, the empirically best fit r value required was 0.723 compared to a value of 0.757 for the 5 mass setup. This reiterates the previous observation that energy dissipation is not strictly as predicted by Housner’s SRM. 4.5 SRM Applied to Rocking on Soil SRM simulations were again conducted to match the column rocking response on the continuous medium. Fig. 9 shows angular displacement time histories of the simulation using an empirically chosen r value of 0.55, and the actual measured response. As expected, the results did not correspond well. While the r value of 0.55 gave approximately the correct peak amplitudes at each cycle, the rocking period was severely elongated. This suggest that the SRM, which only allows for radiation damping at impacts is not suitable for modelling the dynamic behaviour of rocking system in presence of a nonlinear continuous medium rocking interface. Angular Displacement vs Time
0.06
Housner Style ODE Solution (r = 0.55) Test Data (5 Weights)
0.05
Angular Displacement (rad)
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
0
0.5
1
1.5
2
2.5
Time (s)
3
3.5
4
4.5
5
Figure 9. Time history response of SRM simulation and actual test data rocking on soil
4.6 SRM Period Prediction The rocking data for tests on the concrete strong floor and on soil are further compared to the quarter period prediction proposed by the SRM in Eq. 3. Fig.10 shows that Eq. 3 accurately predicted the
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period of rocking when it is in motion. And it should be noted that the quarter period comparison is independent of the r value as they are measured from peak amplitude to impact.
Figure 10. T/4 period comparison of Eq. 3 and the test data (a) on concrete and (b) on soil
Additionally, it can be observed that the quarter periods for the column rocking on soil are noticeably elongated from that predicted by the SRM. This illustrates that the equation of motion from SRM is not adequate for structures rocking on a continuous medium during the rocking phase. 5 CONCLUSIONS This paper confirms the benefits of the use of a rocking mechanism for seismic protection. Furthermore, benefits of a rocking structure being situated on a continuous medium such as soil are demonstrated. It was found that when the restraint conditions were progressively relaxed, from rigidly fixed to rocking on a rigid surface to rocking on a continuous medium, there was a corresponding increase in the equivalent viscous damping as well as an elongation in the natural period of vibration. Experimental data for the sdof column, freely rocking on the rigid concrete strong floor was found to match closely with conventional simple rocking models. However as previous research suggests, there are numerous deficiencies in the current methods of predicting the damping factor. When an apparent coefficient of restitution, r, was used to represent the energy dissipation, a smaller r value than the Housner prediction is required, implying that more energy is dissipated. Additionally, the variation of seismic mass and hence structural properties resulted in little change in this r value. This suggests that the amount of radiation damping is not dependent on such properties as previously anticipated. Although, given the limited experimental data, further experiments are required to confirm this finding. Also, experimental data for the sdof column, freely rocking on a continuous medium showed a dramatic change in the response when compared to that rocking on a rigid concrete surface. The approximate amount of the energy dissipation, expressed as an equivalent damping ratio, was doubled. The principle energy dissipating mechanism shifted from radiation to viscous damping. Attempts to simulate the response using a simple rocking model to this system proved unsuccessful. This clearly demonstrated that the SRM is ineffective when modelling rocking structures on soil. 6 REFERENCES Bruhn, B. and Koch, B. P. (1991). "Heteroclinic Bifurcations and Invariant Manifolds in Rocking Block Dynamics." Z. Naturforsch. A, 46, 481-490. ElGawady, M. A., Q. Ma, J. Butterworth and J. M. Ingham (2006a). The effect of interface material on the dynamic behaviour of free rocking blocks. Eighth U.S. National Conference on Earthquake Engineering (8NCEE), San Francisco, USA. ElGawady, M. A., Q. Ma, J. Butterworth and J. M. Ingham (2006b). Probabilistic analysis of rocking blocks. New Zealand Society for Earthquake Engineering Conference 2006, Napier, New Zealand.
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Gajan, S., B. L. Kutter, J. D. Phalen, T. C. Hutchinson and G. R. Martin (2005). Centrifuge modeling of load-deformation behavior of rocking shallow foundations. Soil Dynamics and Earthquake Engineering 25(7-10), 773-783. Housner, G. W. (1963). Behavior of inverted pendulum structures during earthquakes. Seismological Society of America -- Bulletin 53(2), 403-417. Ma, Q. T., G. D. Wight, J. W. Butterworth and J. M. Ingham (2006). Assessment of current procedures for predicting the in-plane behaviour of controlled rocking walls Eighth U.S. National Conference on Earthquake Engineering (8NCEE), San Francisco, USA. Priestley, M. J. N., S. S. Sritharan, J. R. Conley and S. Pampanin (1999). Preliminary results and conclusions from the PRESSS five-story precast concrete test building. PCI Journal 44(6), 42-67. Shenton, H. W. and N. P. Jones (1991). Base excitation of rigid bodies. I. Formulation. Journal of Engineering Mechanics 117(10), 2286-2306. Wight, G. D., J. M. Ingham and M. KOWALSKY (2004). Shake table testing of post-tensioned concrete masonry walls. New Zealand Concrete Industries Conference, Queenstown, New Zealand. Yim, S. and H. Lin (1991). Nonlinear impact and chaotic response of slender rocking objects. Journal of Engineering Mechanics 117(9), 2079-2100.
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