Seismic Design Considerations for Atlantic Canada
Author: Valerie Latour, EIT and Perry Mitchelmore, P. Eng. of Meco (Mitchelmore Engineering Company Ltd.)
Design professionals are well acquainted with dead, live and environmental loading on structures. Design professionals are also increasingly getting well acquainted with earthquake loading on structures. Despite the fact this familiarity period has lasted for many decades, there is still a gap between analytical knowledge and practice. The most recent edition of the Canadian Dam Association Dam Safety Guidelines (CDA 2007) concurrently provides a significant contribution to the knowledge base available to engineers for practical design applications while making practical design more challenging.
The Guidelines (CDA 2007) present separate descriptions of seismic hazard assessment and seismic analysis. Seismic hazard assessment provides the engineering community with an understanding of the characteristics of seismic hazard, methods of evaluating the hazard and appropriate parameters for seismic evaluation. The seismic analysis is presented separately for geotechnical and structural considerations but are reasonably similar. The challenge for the user is to interpret the intention between the assessment and the analysis.
The Guideline's seismic hazard assessment technical bulletin introduces a new term, "Earthquake Design Ground Motion (EDGM)" to define the parameters for seismic design. The EDGM may include earthquake magnitude and distance, peak ground motion parameters, acceleration response spectra and any potential foundation fault displacements. For dams, the selected parameters in seismic design are the best estimate, or mean, value. The mean value is different from, and generally larger than, the median value available through the National building Code of Canada (NBCC 2005). Canada is fortunate in that dams don't tend to be located on active faults which means the EDGM is nearly always established by a site specific Probabilistic Seismic Hazard Assessment (PSHA), available from the Geological Survey of Canada. The site specific seismic assessment will provide parameters necessary for design, including peak ground acceleration (PGA) and response spectra for different return periods. More detailed information on sources, distance and time histories will require more investigation.
The estimate the EDGM should consider (i) local and regional geotectonic information and (ii) a statistical analysis of historical earthquakes. Using the EDGM, the challenge for designers is to incorporate the parameters from the seismic hazard assessment into available methods of analysis. Methods of analysis tend to be available at two levels; (i) preliminary analysis, which is largely empirical and based on experience in western North American and (ii) detailed analysis, which are highly analytical and based on mathematical models. The Guidelines recommend a progressive approach, starting from a screening level and progressing to full detailed analysis until the design professional is satisfied that the structure is safe.
It sounds fairly simple, but there are peculiarities that the design professional must be aware of when merging the available seismic parameters with the available methods. This paper will attempt to clarify differences between seismic response for concrete gravity dam structures in Eastern North American (ENA) and Western North American (WNA), important considerations in applying empirical methods from difference regions and suggest appropriate methods for simplified analysis.
2. METHODS OF ANALYSIS
In general, the method of seismic analysis is related to the importance of the structure, the consequences of failure, the type of structure and the seismicity of the region. The Guidelines recommend a phased approach for most dams, starting with simple analysis and progressing to more complex methods until satisfied the structure is safe.
2.1 Pseudo-Static Analysis
The pseudo-static seismic analysis is a rigid body, limit equilibrium method of analysis that was traditionally used for design in low seismic risk areas where there was no risk of foundation liquefaction. The method represents dynamic forces by an "equivalent" seismic coefficient multiplied by the structural mass directed horizontally, i.e., orthogonal to the weight vector. Although this method is still used by some design professionals in the preliminary stages of design or safety review, many over simplifications of this analysis method have been realized. These over simplifications can lead to both overly conservative and non-conservative results.
Traditionally, one of the many challenges of the pseudo-static analysis was selection of the seismic coefficient. The tendency was to use the Peak Ground Acceleration (PGA) value, expressed as a percentage of gravity, g, as substitute for the seismic coefficient. This practice was not recommended but there was little guidance on appropriate choices with regards to the link between the seismic parameters used in the pseudo static analysis and equivalent earthquake magnitude. In his Rankine lecture on the subject, Harry Seed (Seed, 1979) used a Newmark analysis procedure to demonstrate that for soils that do not lose more than fifteen (15) percent of their strength during shaking or develop excessive pore pressures, it is only necessary to perform a pseudo-static analysis for seismic coefficient 0.1g for a magnitude 6.5 earthquake and 0.15g for a magnitude 8.6 earthquakes and obtain a factor of safety of 1.15. These benchmarks, intended for embankment dams, were largely adopted in practice for all dams when the pseudo-static method was used.
Specified factor of safety values are required against overturning, sliding and over stressing of the structure - no tension is permitted. In the pseudo-static seismic analyses, the no tension criteria as well as the large factors of safety against overturning and sliding could be achieved due to the unrealistically small earthquake forces being applied to the structure (Chopra and Zhang, 1991).
2.2 Dynamic Analysis
A dynamic analysis is used to consider the dynamic amplification of the earthquake acceleration over the height of a flexible dam responding in its elastic range of motion (Leger and Ftima, 2004). These analyses address many of the over simplifications inherent to the pseudo-static method. In dynamic analyses, the compressibility of the reservoir water as well as the flexible nature of structure and the dynamic dam-reservoir-foundation relationship can be taken into account.
Traditionally, conditions requiring a dynamic analysis to be completed included; dams with active faults directly beneath the dam or reservoir, unusual geometry, large masses near the top of the dam and dams that are higher than fifty feet (15.24m) (USDI, 1987). The International Council on Large Dams (Bulletin 72) suggests that, when assessing a dam in good condition, a dynamic analysis should be applied when the peak ground acceleration (PGA) is greater than 0.25tg and the CEA suggests that a dynamic approach be applied when the pseudo-spectral acceleration at the fundamental period of the dam structure is greater than 0.35g.
Many different linear dynamic analysis methods are available for seismic analysis of dams. These analyses ranger from simplified methods that can be computed by hand to more in-depth analyses that use computer models.
2.2.1 Pseudo-Dynamic Analyses
The pseudo-dynamic seismic analysis is also a rigid body, limit equilibrium method of analysis. However, it uses a response spectral analysis procedure to develop the equivalent static load for rigid body analysis. In the pseudo-dynamic analysis, the dam is treated as a flexible structure and the reservoir is treated as a compressible fluid. The seismic loading depends on the fundamental mode of seismic response of the dam due to the applied horizontal ground motion. The pseudo-dynamic method of analysis is discussed in more detail in section 3.1 of this report.
2.2.2 Linear Dynamic Analyses
For concrete gravity dams, it is reasonable, in most cases, to assume that they dynamic response of the structure will be linear for low to moderate intensity earthquakes (NRC, 1990). A linear analysis is only appropriate for concrete bending in the structure's elastic range of motion. If significant cracking is expected a non-linear analysis is required (Ghrib et al, 1995). Brittle materials such as concrete tend to have significant variations in elasticity depending on the rate of loading -during rapid loading the modulus of elasticity for concrete increases by approximately 25% (NRC, 1990) allowing for a larger range of motion in the elastic range.
For more complex analysis, linear time history analysis method. transient effects of the inertial forces are considered using an accelerogram and the dynamic equations of motion are solved over time (Ghrib et al, 1995). A single impulse of earthquake stress will be generally result in a failure of the structure even if cracking of the concrete is induced because there is not enough time for large displacements (Ghrib et al, 1995). Small cracks that may develop as a result of a single impulse can close before water is able to enter and cause a large displacement.
2.2.3 Non-Linear Analysis
A non-linear analysis is used to quantify motion of a concrete dam structure in its plastic range of motion including cracking; sliding and rocking under seismic ground motions (Leger and Ftima, 2004). A non-linear analysis is required in cases where severe damage is predicted or where the failure condition is approached in the linear analysis and the dynamic behaviour of the dam is drastically changed from the linear response mechanism (NRC, 1990). These non-linear methods generally employ a finite element model and are labour intensive and thus expensive.
3 ANALYSIS FOR ATLANTIC CANADA
When choosing a seismic analysis method for design or evaluation of dams in Atlantic Canada, the design professional must consider that, generally, popular analysis methods were developed for large dams on the west coast of North America. It should not be assumed that a method developed in California can be directly applied to a small concrete gravity dam in Atlantic Canada.
The differences in earthquakes characteristics in Eastern North American (ENA) and Western North America (WNA) are due to differences in the physical properties of the Earth's crust attenuates peak motions very slowly with high peak ground acceleration (PGA) values resulting from relatively small earthquakes over a short duration. In WNA, the crust attenuates the peak ground motions rapidly and high PGA values result from large earthquakes over long durations (Lin and Adams, 2007). What this suggests is that, for example, a seismic event with a PGA of 0.20g. In addition, the high frequency, short period character of ENA earthquakes may, theoretically, be more damaging to small concrete dams with short natural frequencies than WNA earthquakes (James et al, 2004). So while "Big" earthquakes are less common in ENA, designers in Atlantic Canada need to be cautious when choosing an analysis method for seismic design of concrete gravity dams.
The seismic coefficients used in a pseudo-static analysis are a function of earthquake magnitude, which is not a spectral parameter, making traditional pseudo-static analyses susceptible to underestimating the affect that an ENA earthquake will have on a low period structure. These differences indicate that analyses developed for WNA earthquakes, based on prescribed PGA values, will result in non-conservative results if used directly for ENA. By contrast, a pseudo-dynamic Newmark-type sliding block analysis that compares a calculated critical acceleration required to initiate sliding to the expected PGA may result in an overly conservative result in ENA since relatively small, short duration earthquakes can have large PGAs.
Without developing an appropriate method for adjusting peak ground motion parameters to be used in ENA analyses, methods that rely on site specific spectral parameters are more appropriate, as prescribed in the Guidelines. As a result of these observations, the ideal preliminary design method for designers in Atlantic Canada is a simplified pseudo-dynamic analysis that relies on spectral acceleration input parameters.
3.1 Analysis Method - Pseudo Dynamic
For small (less than 15.24 m high), concrete gravity ams in Atlantic Canada, an appropriate preliminary analysis method is the pseudo-dynamic analysis. Using a simplified procedure and considering only the fundamental vibration mode, which is an appropriate assumption for small rigid dams, this method is relatively simple and time effective and thus, cost effective. In addition to the issues raised concerning the applicability of WNA methods in ENA, the benefits of using a pseudo-dynamic analysis over a pseudo-static analysis include the consideration of the affects of the dam-water-foundation interactions and water compressibility - which are not accounted for in the pseudo-static analysis and are considered to play integral roles in the dam's response to earthquake-induced ground motion.
The beam model is often used when analysing concrete gravity dams. The accuracy of this model decreases for higher periods. Small gravity dams are short period structures and their response is mainly in the fundamental mode, making the beam model of analysis appropriate (Ghobarah et al, 1994).
3.2 Design Example
To illustrate the preliminary seismic evaluation of a concrete gravity dam, a design example will be used and the earthquake forces expected will be calculated. The Great Barren Dam, located in Tusket Nova Scotia, will be analysed. The Great Barren Dam was reconstructed in the summer of 2008. The dam consistes of a concrete gravity spillway section founded on bedrock matted with earth embankment sections on each end.
3.2.1 Required Parameters
The site specific mean value ground motion parameters for this site were calculated by the Geological Survey of Canada. The dam safety classification for this structure was determined to be high and therefore the ground motion parameters used are for a probability of 0.000404 p.a. or one in 2500 year return period. The parameters are given then Table 1. Soil Class C, from the NBCC (2005), is used in this case even though the dam is founded on bedrock. This is because the scope of the geotechnical investigation did not include sampling and testing of the bedrock for economic reasons. As a result, the rock could not be confidently classified as Class A, hard rock or Class B, rock so Class C was used to be conservative.
Table 1: Ground motion parameters for NBCC soil class C (very dense soil and soft rock).
Values for a probability of 0.000404 p.a.
The pseudo-dynamic analysis followed is Fenves and Chopra's simplified analysis for concrete gravity dams. This analysis is a simplified version of the general dynamic analysis procedure that includes the effects of the dam-water interaction and compressibility of water as well as dam-foundation rock and reservoir bottom interactions. The maximum response of the dam is estimated directly from the earthquake design spectrum (Fenves and Chopra, 1986). This analysis was chosen for its ease of computation and consideration of key parameters making it both relatively reliable and cost effective. It will be assumed the dam will respond in the fundamental mode and therefore only the stresses as a result of the fundamental vibration mode will be considered. For the purpose of this analysis, only the concrete gravity section will be considered. Tables from Fenves and Chopra, 1986, that were used to obtain design parameters are included in Appendix A for reference.
The input parameters required for this analysis are presented in Table 2. The reservoir water level is considered to be at the top of the spillway section (H/Hs = 1.0).
Table 2: Input parameters for the Great Barren Dam
*Site specific characterisitics of the Great Barren Dam
**Material specific parameters
The parameter values that are not site or material specific where recommended for the analyses by Fenves and Chopra, 1986. The wave reflection coefficient, hysteretic damping coefficient and viscous damping ratio of concrete values were recommended for cases, such the design example, where the actual values are not known. The recommended values for the wave reflection coefficient ranged from 0.75 to 1.0, depending on the predicted sediment deposit at the toe of the dam where 1.0 represents little to no deposited sediment (Fenves and Chopra, 1986).
3.2.2 Computation of Equivalent Earthquake Force
All computations were carried out using imperial units and the resultant calculated forces were converted to metric units. The fundamental vibration period of the dam, founded on a rigid bedrock foundation, including the influence of the reservoir,
Where Rr is the period lengthening ratio, a factor dependent on the modulus of elasticity of the concrete the depth of water in the reservoir and the absorptiveness of the reservoir bottom. Rf is the period lengthening ratio due to dam-foundation rock interaction and is a function of the modulus of elasticity of the concrete and the foundation rock. These actor values are selected from a table given in Fenves and Chopra, 1986.
The period ratio, Rw is calculated next. For values of Rw < 0.5, the effects of water compressibility are negligible.
Where T1'=4H/C. Since Rw = 0.92 > 0.5, the affects of water compressibility are not negligible and assuming the reservoir water to be incompressible is not a valid assumption. The fundamental vibration period of the dam considering the influence of foundation rock flexibility the influence of the reservoir, The damping ratio of the dam is calculated from equation (4). is a function of the damping ratio of the dam on rigid rock foundation, the added damping ratio due to hydrodynamic effects and the added damping due to dam-foundation rock interaction, a factor dependent on the modulus of elasticity of the concrete the depth of water in the reservoir and the absorptiveness of the reservoir bottom is a function of the modulus of elasticity of the concrete and the foundation rock. These factor values are selected from tables given in Fenves and Chopra, 1986.
The generalized mass, is computed from equation (5) where is generalized mass force coefficient. The generalized earthquake coefficient, is computed from equation (6).
Where Ws is the weight of the structure, Fst = wH2/2 is the total hydrostatic force on the dam and Ap is the integral of the function wgp(ŷ)/wH.
The equivalent lateral earthquake force associated with the fundamental vibration mode, f1(y) varies over the height of the structure.
Where ws(y) is the weight of the dam at height y and is the function of the fundamental vibration mode shape. A summary of calculated values as well as design parameters selected for use in the analysis is given in Table 3.
The equivalent lateral earthquake force, calculated over the height of the Great Barren Dam is illustrated in Figure 1. This force distribution can be used in a structural analysis or global stability analysis to estimate the performance of the dam under seismic loading.
The spectral acceleration function, used to calculate f1(y) was evaluated at a period of 0.2; the lowest period reported in the seismic hazard calculation for a 5% damping ratio. The fundamental vibration period of the dam was found in equation (1) is 0.012 seconds. The damping ratio that is to be applied to this spectral acceleration, as calculated in equation (4) is 7.5%.
There are several methods available to assess the seismic response of a concrete gravity dam. The traditional pseudo-static method, which relies on peak ground acceleration, does not provide reliable results. More comprehensive methods are available that use response spectral analysis parameters and provide more reliable results. The response spectral methods are consistent with the latest edition of the National Building Code of Canada (NBCC 2005) and are amenable to probabilistic assessment, as prescribed by the Guidelines (CDA 2007). Society would expect dam engineers to use at least an equivalent method for dams as for buildings, particularly for high consequence dams.
Some caution is required when adopting spectral analysis methods. Small concrete gravity dams are short period structures by nature and the applicability of substituting the spectral acceleration value for the lowest period available, Sa(0.2), we can see an obvious trend to increasing spectral accelerations for Atlantic Canada compared with western Canada.
By linear extrapolation, we can discern that the spectral response will be heightened for smaller period structures, such as dams, and that the spectral response for a period equal to 0.2 may be under conservative. A comparison of mean values from NBCC for median values is presented in Figure 2.
For the lowest available acceleration parameter to be appropriate for use in design calculations, concrete gravity dams, which are chaacteristically low period structures, have to be quite large. Using equations (1) and (2), and setting and solving for Hs, a conventional concrete gravity dam has to be sixty-six (66) meters high to have a fundamental vibration period of 0.2 s. This suggests that for the majority of concrete gravity dams in Atlantic Canada, using the parameters available, the spectral acceleration is underestimated.
Seismic design considerations for Atlantic Canada differ from those in Western Canada for a number of reaons
- Because of differences in crust attenuation, seismic events tend to be high frequency, low duration events in Atlantic Canada.
- Further reserach and analysis is required to develop spectral acceleration parameters for periods below 0.2 at the National Reserach Council to be of assistance for dams. The linear extrapolations assumption needs to be verified.
- The damage "potential" of a seismic event in Atlantic Canada is equivalent to a similar event in Western Canada at lower peak ground acceleration (PGA) values.
- As most seismic evaluation methods were developed based on WNA experience, different prescribed criteria should be used for ENA. This is particularly the case for the pseduo-static method, which will either over estiamte or under estimate a response depending on how the method is applied.
- Seismic evaluation for concrete gravity dams in ENA should use site specific resonse spectral analysis gathered from a Probabilistic Seismic Hazard Assessment (PSHA).
- A screning approach is suggested, starting with pseudo-dynamic assessment and progressing to full linear and non-linear methods, until satisfied the structure is safe or modifications to make the dam safe are understood.
For small concrete gravity dams, the pseudo-dynamic assessment represents a cost effective method of reliably assessing seismic safety of the dam and will typically be all the assessment needed for ENA.
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