For the verification we used three different sets of data containing measured time histories of three different earthquakes • Friuli Italy 1976 (Station Forgaria Cornino, Station San Rocco)
• Coalinga California (USA) aftershock 1983
• The earthquakes used by Keintzel [3] and Hoeflich [4]
These earthquakes are well documented. All relevant seismological, geophysical and geological data are well known. With respect to magnitude and epicentral distance they are comparable to German conditions. The focal parameters and the geological data of the layers crossed by the seismic waves were supplied by the Geophysical Institute of the
University of Stuttgart.
The first data set was chosen from the numerous registered time series of the Friuli Italy 1976 earthquake. Only stations with short epicentral distances, where the direct S-waves govern the character of the ground motions, were selected. Furthermore, earthquakes with magnitudes of M>5.0 were preferred. The stations Forgaria Cornino and San Rocco were chosen, because the subsoil of Forgaria Cornino consists of layers of soft soil, whereas San Rocco is built on rock. The geological data can be found in [5]. The parameters are presented in tables 3 and 4. The complete set of input data used for the simulation of the artificial earthquakes is listed in tab. 1 to 5.
The second data set contains selected accelerograms of the Coalinga earthquake (Jul. 22. 1983 2:39 UTC Tab. 6). The generalised geological section and the focal parameter can be seen in tables 6 and 7. Because of the significance, sites with the short epicentral distances of 5, 10 and 17km were chosen.
The geophysical parameters of the earthquakes used by Keintzel and Hoeflich are listed in [12]. These accelerograms of older American earthquakes were modified in the frequency content and scaled in the Fourier spectrum
to correspond to German conditions. The detailed procedure is described in [3]. For the simulations a geological section and focal parameters typical for a site in the Swabian Jura with an epicentral distance of 7, 5km were used. The
simulations made with this third data set are not used for verification. They were only used to check the computer code in comparison to the results by Keintzel and Hoeflich.
Linear elastic response spectra
The main purpose of the research project was to examine statistically the parameters of the measured and simulated earthquake time histories. The simpliest way is to use linear elastic response spectra. We started with the maxima of the displacements, velocities and accelerations. However, it is well known that the maxima of the response do not always describe the seismic load significantly. It is also well known that the duration of the strong motion phase is very important. Therefore also energetic expressions were analysed. Starting with the differential equation
(9) where
ω eigenfrequency
D damping
acceleration of the soil
it is possible to obtain an equation of equilibrium of energy by multiplying Eq.(9) with the velocity and integrating
(10) or Ekin+Evis+Eel=Einp (11) where (12) (13)
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(14)
(15) Time series of these integrals can be seen in the addendum A 2 of [12] for different eigenvalues of the single degree of freedom oscillator. Maxima shown in a diagram are similar to the well known response spectra. For the verification of the simulation procedure we used
max a maximum acceleration
max ν maximum velocity
max d maximum displacement
max Einp maximum input energy An example can be seen in Fig. 6.
Elastic multi degree of freedom systems
Linear response spectra show the reaction of a single degree of freedom system due to base acceleration. For multi degree of freedom systems the mode superposition method can be used. Hence, all the results of the previous chapter are also valid for multi degree of freedom systems. Therefore it is not necessary to treat this problem separately.
Nonlinear response spectra
For the nonlinear system of Fig. 2 the same analysis is made as for linear elastic response spectra. The results are similar spectra but they reflect the characteristics of the nonlinear oscillator. In the differential equation (9) the term with the spring constant is now replaced by the elastic plastic resistance of the system.
(16) where
with
Rel=max R=xelk=const.
The yield load was defined by 80% of the ultimate load and the critical value xel by the elastic displacement at that load. Possible strain hardening was neglected as a horizontal yield plateau was assumed. It was appropriate for our analysis to divide the potential energy in two parts as shown by the following equations
(17)
(18)
xpl(t)=x(t)−xel (19)
The elastic energy Eel is restored to the system and is available for future oscillations. The hysteretic energy Ehyst is dissipated and therefore not available for future oscillations. The response is reduced.
The following values are compared for verification. max a maximum acceleration max ν maximum velocity max d maximum displacement
max dpl maximum plastic displacement max Einp maximum input energy
max duc maximum ductility factor max Ehys maximum hysteretic energy max Evis maximum dissipated energy The detailed results are given in tab. 13 and 14 of [12].
VERIFICATION (BUILDINGS)
An engineer does not ask for simulated earthquake time histories fitting given spectra. He is interested in other questions, e. g.: Is the nonlinear performance of a building with respect to its plastic deformations in critical regions comparable for real (measured) and artificial (simulated) earthquakes? To obtain relevant results we have chosen two types of typical structures consisting of moment resisting frames or of shear walls with 5, 10 or 20 stories. We used only computer codes with reliable and experimentally verified nonlinear models, as for example, the Takeda model.
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Moment resisting frames
Regular and symmetric frame type structures with two bays and five or ten storeys (Fig. 3) were analysed. These frames had already been designed by Hoeflich [4] for a distributed load of 10 kN/m2. The earthquake loads corresponded to the earthquake zone 4 of the German code DIN 4149 [8].
The German earthquake standard DIN 4149 sets two limits for the axial load in columns
(n≤0.23 or n≤0.5) (20)
(21) where
n nondimensional axial load N axial load in the column Ac concrete cross section Rc nominal concrete strength
n≤0.23 can be used without restrictions while n≤0.5 is only allowed in connection with special requirements for a higher ductility. 4 types of frames were obtained. The sections are given by tables 15 and 16 of [12]. In tables 17 and 18 of [12] the three lowest eigenvalues are presented. The frames were loaded with the base accelerations of the measured and simulated earthquake time histories.
For all relevant sections, the moment curvature relation for the different reinforcement was calculated with the computer code ZAEH 1 [3]. The constitutive laws for concrete and reinforcing steel were assumed according to German standards. Strain hardening was not taken into account.
The nonlinear analysis was made with the computer code SAKE [9] especially developed for moment resisting plane frames with degrading stiffness according to Takeda [10] as shown in Fig. 5. Damping is considered as
Rayleigh-damping so that the damping of the two lowest modes is D=0.05. P-∆ -effect, i.e. the increase of moments due to large deflections is taken into account.
For the verification, we compared the following values: • maximum storey acceleration
• maximum storey deflection
• maximum bending moment at the base of columns • maximum base shear
• maximum ductility factors required for girders, inner and outer columns.
The results can be seen in tables 19 to 22 of [12] and for one example in Fig. 7. Generally, the structural response of real and simulated earthquakes corresponds very well to each other.
Shear wall structures
Only two-dimensional systems with uncoupled shear walls and 5, 10 and 20 stories were considered (Fig. 4). Due to the slenderness of the walls the predominant part of the elastic and plastic deflections are caused by bending deformations. However, shear deformation was included in the analysis. It was assumed that the shear reinforcement remains in the elastic range so that only linear elastic shear deformations need to be considered. This assumption can always be realized in design by using adequate shear reinforcement.
Maximum stresses and strains of slender shear walls under horizontal load always occur near the base. Therefore only the overturning moment with earthquake loads according to the German standard DIN 4149 was calculated. The systems shown in Fig. 4 with constant section and mass distribution were assumed. The distributed load of 10kN/m2 (including dead and life load) for one storey yields to a storey mass of 60kN s2/m. The stiffness of the wall was determined so that periods of 0.4s, 0.8s, 1.6s and 2.4s were obtained.
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For the verification, the following values were considered: • required ductility factor at the base
• accumulated plastic hinge rotation at the base.
The analysis was made with the nonlinear finite element program DRAIN-2D [11] with the degrading stiffness as in the Takeda model (Fig. 5). Damping was assumed as Rayleigh damping corresponding to a damping ratio D=0.05 in the first two modes.
The results can be seen in tables 23 to 25 of [12] or in Fig. 8. It is obvious that the stresses and strains in the structure of most of the real earthquakes agree well with the simulated ones. But it can also be seen that sometimes the results are over- or underestimated. This tendency for poorer results is not surprising. Contrary to frame type structures, the complete plastic deformation is concentrated in one single hinge at the base of the shear wall. In this hinge all
inaccuracies resulting from the basic assumptions of the earthquake simulation up to the approximations in the structure, are expressed in an offset from the true value. These results show very clearly the equalizing and compensating effect of statically indeterminate structures where the plastic deformations are distributed over many plastic zones in the structure.
CONCLUSION
Nowadays a great number of measured strong motion data of earthquakes exist. They are collected in databases.
However, these databases must contain gaps. With our site dependent simulation model it is possible to close these gaps in two different ways. First, all geometrical parameters concerning properties of soil or rock layers, epicentral distance, focal depth can be chosen arbitrarily. Second, it is possible to interpolate over all magnitudes up to about Ms=6.5. Further on, it is surely possible to extrapolate up to Ms=7.0. So it is possible to simulate the earthquakes with the highest damage potential. However, the model does not allow to simulate earthquakes with
• surface faulting because of the importance of the neglected surface waves
• magnitudes higher than Ms=6.5–7.0 because of the enormous dimensions of the focus, which cannot be assumed as a point source.
Earthquakes of this type are not important for German conditions. Therefore it can be safely assumed that the proposed simulation model covers nearly all cases occurring in Germany.
In detail, we found the following degree of agreement between real and simulated earthquakes:
● linear response spectra: good agreement
● nonlinear response spectra: relatively good agreement ● frame type buildings: good agreement
● shear wall buildings
– mean value: agreement – maxima: poorer agreement
If the plastic hinges are distributed more or less uniformly over the structural system, the response of real buildings can be well predicted.
The advantages of this method can be summarized as follows:
● Generally, good results can be expected ● Input data is relatively simple
● Individual site data such as geological layers, subsoil properties, focal depth and epicentral distance can be prescribed
arbitrarily
● The simulation covers earthquake magnitudes from microseisms up to M≥6.5 and includes the possibility to
extrapolate up to
● All current data normally derived from earthquake measurements, such as accelerations, velocities, displacements,
energies, different types of magnitudes, duration of the strong motion phase and all the related statistical data can be calculated.
The disadvantages of the method in the present version are the following:
● The influence of surface waves is not taken into account. The method is not appropriate for earthquakes with surface
faulting or for great epicentral distances. In principle, however, it is also possible to include surface waves.
● The model is completely linear. Nonlinear effects are not regarded but are approximately considered by equivalent
linearisation.
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It should be mentioned that the proposed simulation model is not a phenomenological but a physical one. This means that the focal process is physically modelled in a statistical sense. Also, the energy transmission by waves through the geological layers up to the surface is physically modelled with all reflections and refractions at the layer interfaces. The seismic moment is used for scaling on a geophysical basis.
ACKNOWLEDGEMENT
The scientific support of Dr. F.Scherbaum and Prof. G.Schneider, University of Stuttgart, in the computer program arrangement and in selecting data for model verification is highly appreciated. This work was sponsored in parts by DFG (Deutsche Forschungsgemeinschaft) and DAAD (Deutscher Akademischer Austauschdienst).
REFERENCES
1. Schneider, G., Scherbaum, F.: DFG-Forschungsvorhaben “Erdbebengrundlagen”, Nr. 354/23, Bericht zum 1. Nov. 1982
2. Boore, D.: Stochastic Simulation of High-Frequency Ground Motions Based on Seismological Models of the Radiated Spectra, Bull. Seism. Soc. Am.. 73/61, pp. 1865–1894, December 1983
3. Keintzel, E.: Zähigkeitskriterien für Stahlbetonhochbauten in deutschen Erdbebengebieten, Dissertation, Universität Karlsruhe (TH), 1981
4. Hoeflich, S.G.: Nichtlineares Verhalten von Stahlbetonbauten unter Erdbebenbelastung, Dissertation, Universität Karlsruhe (TH), 1983
5. Scherer, R.J., Schueller, G.I.: Friuli Earthquake Sequence of 1976. Records and Power Specta of Corrected and Integrated Strong Motion Earthquake Data, Insbruck-Munich, May 1985
6. Housner, G.W.: Intensity of Ground Shaking Near the Causative Fault, Proceed, III WCEE, pp. 81–94, 1965 7. Sargoni, G.R., Hart, G.C.: Simulation of Artificial Earthquakes, Earthquake Eng. and Structural Dynamics, Vol. 2, 1974 pp. 249–267
8. DIN 4149 Bauten in deutschen Erdbebengebieten Lastannahmen, Bemessung und Ausführung üblicher Hochbauten (German earthquake standard).
9. Otani, S.: SAKE, a Computer Program for Inelastic Response of R/C Frames to Earthquakes, University of Illinois, Urbana, 1974
10. Takeda, T., Sozen, M.A., Nielsen, N.S.: Reinforced Concrete Response to Simulated Earthquakes, ASCE, Journal of the Structural Division, Vol. 96, 1970
11. Kanaan, A.E., Powell, G.H.: DRAIN-2D, A General Purpose Computer Programm for Dynamic Analysis of Inelastic Plane Structures. EERC, Rep. 73–6 and 73–22, University of California. Berkeley, California, 1973
12. Eibl, J., Henseleit, 0. and Kostow, M.: DFG-Forschungsvorhaben “Erdbebengrundlagen” Nr. Mu 354/23 Teil 2. Bericht zum 31. Dezember 1986
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Tab. 1. Friuli Earthquake 1976, Station Forgaria Cornino
N time ML R[km] h[km] comp. max. a [g/10]
1 76.09.11 5.3 20.6 6 NS 1.0000 16.31.10 WE 1.1429 2 76.09.11 5.6 19.4 6 NS 1.3131 16.35.01 WE 2.3382 3 76.09.15 5.9 15.9 5 NS 2.6367 03.15.19 WE 2.1934 4 76.09.15 6.0 15.8 7 NS 3.5404 09.21.18 WE 3.3505 5 77.09.16 5.3 6.1 8 NS 2.4557 23.48.07 WE 2.0139 6 76.05.11 5.0 7.6 6 NS 1.9192 22.43.60 WE 3.1185
Tab. 2. Friuli Earthquake 1976, Station San Rocco
N time ML R[km] h[km] comp. max. a [g/10]
1 76.09.11 5.3 20.6 6 NS 0.3403 16.31.10 WE 1.7151 2 76.09.11 5.6 19.4 6 NS 0.8024 16.35.01 WE 0.9555 3 76.09.15 5.9 15.9 5 NS 0.6045 03.15.19 WE 1.3556 4 76.09.15 6.0 15.8 7 NS 1.3092 09.21.18 WE 2.5106
Tab. 3. Geological Cross Section, Forgaria-Cornino
layer no. layer thickness S-wave velocity density quality-factor
[m] [m/s] [kg/m3] σ σ σ σ 1 5 1 200 50 1800 100 20 5 2 21 2 600 100 2100 100 50 5 3 500 50 900 100 2100 100 100 10 4* 9000 — 2500 — 2500 — 200 — *half space
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Tab. 4. Geological Cross Section, San Rocco
layer no. layer thickness S-wave velocity density quality-factor
[m] [m/s] [kg/m3]
1 30 5 600 100 2000 200 50 5
2 2000 200 2500 200 2200 100 100 10
3* 9000 — 3000 — 2400 — 200 —
* half space
Tab. 5. Focal Parameter, Friuli Earthquake
N earthquake date/time focal depth radiation pattern stress drop moment magnitude
[km] [MPa] 1 76.09.11 16.31.10 6 0.7 0.1 1.5 0.5 5.2 0.25 2 76.09.11 16.35.01 6 0.7 0.1 1.7 0.5 5.5 0.25 3 76.09.15 03.15.19 5 0.7 0.1 1.5 0.5 5.85 0.25 4 76.09.15 09.21.18 7 0.7 0.1 1.5 0.5 6.0 0.25 5 77.09.16 23.48.07 8 0.7 0.1 1.5 0.5 5.2 0.25 6 76.05.11 22.43.60 6 0.7 0.1 1.8 0.5 5.0 0.15
Tab. 6. Dataset Coalinga. Characteristics of the Accelerograms
no. station distance
[km]
comp. max. acceler. [cm/s2]
max. velocity [cm/s]
max. displacement [cm]
1 Coalinga, Burnett Construction 11.3 360 330.48 17.54 1.35
270 −251.58 15.70 2.39
2 Coalinga, Oil city 5.0 360 −454.72 34.13 −9.45
270 −920.76 38.20 −6.22
3 Coalinga, Oil Fields, Fire Station, Freef. 9.4 360 189.45 −15.88 −3.50
270 −212.05 −16.78 3.39
4 Coalinga, Oil Fields, Fire Station, Pad 9.4 360 215.64 −16.76 −3.71
270 −209.95 −16.91 3.52
5 Coalinga Palmer Avenue 10.0 360 312.52 −21.32 −3.42
270 283.94 −13.20 2.22
6 Pleasant Valley, Pump Plant, Basement 17.40 135 −141.38 5.61 −1.63
095 430.65 −25.93 4.59
7 Pleasant Valley, Pump Plant, Basement 17.40 360 419.99 −21.84 −2.99
270 −230.55 19.44 −4.12
8 Coalinga, Skunke Hollow 11.10 360 229.84 14.86 −3.65
270 365.33 16.23 −3.27
9 Coalinga, Transmitter Hill 6.90 360 1145.73 −46.82 −4.40
270 −834.62 46.58 6.15
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Tab. 7. Geological Cross Section, Coalinga, Californien
layer no. layer thickness S-wave velocity density quality-factor
[m] [m/s] [kg/m3] 1 500 100 600 100 1800 100 50 5 2 500 100 850 200 1800 100 50 5 3 1150 150 1750 200 2100 100 150 50 4 850 150 2050 200 2200 100 150 50 5 1000 100 2300 200 2300 100 200 50 6 900 100 2650 200 2300 100 200 50 7 2400 — 2800 — 2600 — 200 —
Fig. 1. Flow chart of computer code “SIMUL”
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Fig. 2. Plane frames for numerical comparisons with computer code SAKE
Fig. 3. Numerical model for SAKE
Fig. 4. Deformation and numerical model of a horizontally loaded shear wall
Fig. 5. Determination of accumulated plastic deformation
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Fig. 6. FRIULI EARTHQUAKE, FORGARIA CORNIND RESPONSE SPECTRA, DAMPING 0. AND 0.10 ———REAL DATA