Dynamic Forces in Slabs of Concrete-Faced
Assistant Professor of Civil & Environmental Engineering,
A simplified analytical method is presented for determining the dynamic strains and stresses in slabs of concrete-face rockfill dams. Results of the method compare favorably with those obtained using sophisticated finite-element formulations.
KEYWORDS: Concrete-faced, Rockfill dams, natural period, dynamics
Presently, there is little analytical or observational evidence on the behavior of concrete slabs that cover the upstream slope of Concrete-Faced Rockfill (CFR) dams, during strong seismic shaking. The choice of slab thickness and steel reinforcement is based solely on precedent, and performance under static loads is the only consideration. Filling this gap, the author has recently performed a theoretical study (Uddin & Gazetas 1995) that explored the potential consequences of strong seismic shaking for slabs of a typical 100 m-tall dam, using state-of-the-art finite-element (FE) analyses. Particular attention was given to properly modeling the soil-slab interface, assuming a tensionless contact obeying Coulomb's friction law. Evidently, such analyses cannot be performed easily in the design office. In this article we develop a simple analytical method, to compute slab stresses and strains. The motivation for the assumptions on which the method is based has come from the results of the FE studies. Despite its simplicity the method is shown to be in accord with the FE method.
With reference to the vibration in the y direction (Figure 1), the following simplifying assumptions are made based on the results from the finite element studies.
Figure 1. Schematic Illustration of Effects of Rocking Oscillation of a Concrete Face Rockfill (CFR) Dam: (a) Modeling of Dam Geometry with Forces Acting on Elemental Section, and (b) Coordinate Systems for Face Slab
1. The vibration characteristics of the CFR dams can be estimated with conventional methods, ignoring the slab (Uddin 1992).
2. The dynamic strains and stresses induced in the concrete face slab are due only to the rocking ("bending"-type) mode of vibrations. Indeed, the FE studies (Uddin 1992) showed that neither shearing deformations (which are most important for the overall response of the dam) nor other types of deformation produce any significant stress in the slab.
3. The dynamic response of the dam is not influenced by the presence of the slab. Instead, the slab is considered as just the outer "skin" of the dam, following it in its deformation. (Of course, the slab plays a very important role in keeping the dam unsaturated, increasing the effective confining pressures, and thereby enhancing the stiffness of the dam. These effects are must be and taken in to account.) Continuity is assumed between the slab face and the plinth where there is generally a construction joint.
4. The dam is observed to vibrate as a flexural cantilever beam, obeying Bernoulli's hypothesis that plane horizontal sections rotate but remain plane (Uddin & Gazetas 1994).
5. No reservoir hydrodynamic pressures are considered as they have been shown by many researchers to be relatively insignificant due to the mild slopes of the face (typically 1V to 1.5H).
Denoting by u(z,t) the lateral deformation due to bending, the equations of motion in rocking vibration are given by
where ρ = mass density; α = B(z)/z where B(z) = length of the horizontal section at depth z (typically, α=3); and I=α3z3/12 = moment of area of this section. Making use of the stress-displacement relations that are consistent with the aforesaid assumptions and eliminating M and T, leads to the following governing equation for free vibration in rocking:
where E = 2G(1+v) = Young's modulus, G = shear modulus and v = Poisson's ratio of the rockfill material. Setting ζ = z/H and substituting u=U(ζ)sinωt, where U(ζ) is the vibration amplitude, reduces Eq.2 to
where A = E/12H2; B = ρω2/α2.
If we use the power series
as solution, the coefficients an can be determined from the boundary conditions:
(i) M = 0 and T = 0 at ζ = 0 ; and
(ii) U = 0and dU/dζ = 0 at ζ=1
and the form of free vibration may be found. After performing calculation, it is found that the period of free vibration can be solved from the roots of the following equation
where B/A = (12ρH2ω2)/(α2E) = m is used.
In this case the coefficients of the terms higher order than m4 are omitted. Since the minimum root of the above equation is m0 = 28.205, the period TB of the fundamental mode in bending vibration can be written as:
which is identical to the classical Kirchhoff (1850) solution. Substituting a Poisson’s ratio v=0.30,
Recall that the fundamental period in shear vibration is (Gazetas & Dakoulas 1992):
Evidently, TB is smaller by a factor of about 3 than Ts, as expected for a stiffer behavior.
The alternating elongation and shortening of the outer “skin” of the dam in essence produces the strains in the slab. Naturally, this is a displacement-controlled process (whereas by contrast the sliding wedge deformations in dams are acceleration-controlled). With this in mind, and noting that the second natural bending period is about only 1/10 of TB (or equivalently that the 2nd frequency is about 10 times fB), higher bending frequencies (which contribute significantly only in acceleration response) have a negligible effect on slab deformation, even with tall (relatively flexible) dams. It is, therefore, sufficient to consider only the first bending mode (displacement mode) in calculating slab response.
The bending displacement shape i.e., the form of fundamental mode can be obtained as
Hence, u(z,t) is harmonic with frequency ωB = 2π/TB :
where ωd = ωB Ö(1-ξB 2)» ωB for small values of ξB, and
PB = the first bending mode participation factor
and SA(t) is the acceleration response of a single degree of freedom system with frequency, ωd and ξB= damping ratio in the rockfill.
Dynamic Axial Strains and Stresses in the Face Slab
Assuming the dam as a composite rotating beam (Figure 1b) the axial strain of the slab εaxial can be related to the slope of the u-displacement curve as follows:
where y is the distance from the dam axis (Figure 1a). Substituting
u by Eq. (7), α' = α/2, y = α'z = α'ζH, and ζ=z/H=1-X/L where X = the distance along the slab from the ground and L = the length of the slab, one can express the slab strain as:
in which ωd = ωB Ö(1-ξB 2)» ωB for small values of ξB.
It is of interest to focus on the slab strain participation defined as the product of the participation factor times the bending shape: ΦB = ΦB(X/L) = PB (1-X/L) │d2UB(ζ)/dζ2│ζ = (1-X/L). If the variation of the ΦB value throughout the slab is plotted, as a function of the normalized distance X/L, where L is the length of the slab, it will be noticed that │ΦB(X/L)│maxoccursat about 2/3 distance (X/L » 0.6) from the ground.
The expression for the maximum axial strain for the slab may be written as:
where SA is the spectral acceleration corresponding to the bending
Application of the Analytical Method
To explain the application and evaluate the accuracy of the developed simplified method, the response of a 100-m high CFR dam, for which a solution has been obtained by the FE method (Uddin and Gazetas 1995), is calculated herein.
In the FE solution the maximum shear modulus was obtained as a function of the square root of the effective confining pressure; therefore, a weighted average based on values from all elements is used for the homogeneous section analyzed by the approximate method. The properties of the CFR dam are as follows: H = 100 m; up-and down-stream face slopes, α = 3 (or slope angle β = 33.60); mass density ρ = 2.1 t/m3, average Gmax = 1.44 GPa; Poisson's ratio n = 0.30, slab thickness t=0.40 m, the crest width 10 m wide; and reservoir at its maximum level, 90 m. Five acceleration records for five historic earthquakes are used as the input ground motions for the FE analyses (Uddin & Gazetas 1995). Based on the peak ground acceleration (PGA) and distance from the source these five records are divided into two groups:
l Very Strong Motion (PGA ³ 0.6 g): GIC record in the San Salvador (October, 1986) earthquake and PV record in the Coalinga (May, 1983) earthquake
l Moderately Strong Motion (PGA » 0.4 g): El Centro A1 station in the Imperial Valley (October, 1979) earthquake, NU record in the Nicaragua (March, 1973) earthquake and AD record in the Morgan Hill (April, 1984) earthquake.
Sample calculations are shown here for one record from each group. For a consistent comparison, G/Gmax obtained in terms of the average shear strain amplitude in the final iteration of the FE analyses are used to obtain bending frequencies. Young's modulus of the slab is taken to be the same as the one used in FE analysis (= 22000 MPa).
Moderately Strong Motion: Nicaragua Record.
(G/Gmax)avg = 0.25, and (Eq. 4) TB= 2.5 [2142/(0.25 x 1.44x109)]0.5 (100/3) = 0.2 s, for which the spectral acceleration from Figure :2: SaB » 8 m/s2. Eq. 11 then gives maximum slab stress σmax »3.4MPa.
Figure 2. Response Spectra for (a) Moderately Strong Earthquake and (b) Very Strong Earthquake Records
Very Strong Motion: San Salvador record.
(G/Gmax)avg = 0.15, and (Eq. 4) TB= 2.5 [2142/(0.15 x 1.44x109)]0.5 (100/3) = 0.26 s, for which the spectral acceleration from Fig. 3 : SaB » 20 m/s2. Eq. 11 then gives maximum slab stress σmax »13.4MPa.
These values are indeed very similar to that of the FE solution (Figure 3). In particular, the method reproduced quite closely the FE results with "allowing slippage" interface. The simplifying method leads to an under-prediction of merely 2% for the maximum peak axial stress in the slab. It can be explained by that fact that only fundamental rocking mode is considered and contributions of higher modes are not included.
Figure 3. “FE Analysis” Results for
(a) Moderately Strong Nicaragua 1973, NU Record; and
(b) Very strong San Salvador 1986, GIC Record
A simple yet rational approach to the design of concrete face slab for the CFR dam under earthquake loading has been described herein. The method is based on the results from FE studies. The method is an approximate one and involves a number of simplifying assumptions that may lead to reasonable result.
Finally, the method has been applied to an 100 m high CFR dam subjected to historical records where approximate peak slab response are in good accord with the FE results. Provided that one uses realistic elastic parameters depending on the excitations, the presented method can be readily applied by engineers to estimate slab strains and stresses in a face slab by the hand calculation method given.
Gazetas, G. and Dakoulas, P. (1992). " Seismic analysis and design of rockfill dams: state of the art". Soil Dynamics and earthquake Engineering, Vol. II, pp. 27-61
Hatanaka, M., (1955). "Fundamental Considerations on the Earthquake Resistant Properties of the Earth Dam." Bull. No. 11, Disast. Prev. Res. Inst., Kyoto Univ.
Uddin, N., (1992). "Seismic Analysis of Earth-core and Concrete- face Rockfill Dams." Ph.D Thesis, State University of New York, Buffalo.
Uddin, N. and Gazetas, G. (1994). "Seismic Analysis of Concrete- face Rockfill Dams." CEE Report, State University of New York, Buffalo.
Uddin, N. and Gazetas, G. (1995). " Dynamic response of Concrete-Face Rockfill Dams to Strong Seismic Excitation.” Journal of the Geotechnical Engineering. ASCE. Vol. 121, No. 2.
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