Dynamic in-plane transversal normal stresses in the concrete face of CFRD

Abstract

Severe earthquakes can induce damages to Concrete Face Rockfill Dams (CFRDs) such as concrete cracking and joint’s water stops distressing where high in-plane transversal normal stresses develop. Although these damages rarely jeopardize the dam safety, they cause large water reservoir leakages that hinder the dam functioning. This issue can be addressed using well know numerical methods; however, given the wide range of parameters involved, it would seem appropriate to develop a simple yet reliable procedure to get a close understanding how their interaction affects the CFRD’s overall behavior. Accordingly, once the physics of the problem is better understood one can proceed to perform a detailed design of the various components of the dam. To this end an easy-to-use procedure that accounts for the dam height effects, valley narrowness, valley slopes, width of concrete slabs and seismic excitation characteristics was developed. The procedure is the dynamic complement of a method recently developed to evaluate in-plane transversal normal stresses in the concrete face of CFRD’s due to dam reservoir filling [¹]. Using these two procedures in a sequential manner, it is possible to define the concrete slab in-plane normal stresses induced by the reservoir filling and the action of orthogonal horizontal seismic excitations acting at the same time upstream-downstream and cross river. Both procedures were developed from a data base generated using nonlinear static and dynamic three-dimensional numerical analyses on the same group of CFRD’s. Then, the results were interpreted with the Buckingham Pi theorem and various relationships were developed. In the above reference, the method to evaluate the concrete face in-plane transversal normal stresses caused by the first reservoir filling was reported. In this paper, the seismic procedure is first developed and then through an example the whole method (dam construction, reservoir filling plus seismic loading) of analysis is assessed.

Keywords

CFR dams / dynamic analysis / in-plane normal stresses / concrete face

Introduction

Nowadays there are very few experiences in the seismic behavior of CFRDs. Most of these structures have been built on very low seismicity countries such as Australia and Brazil and those built in highly seismic zones in Mexico have not being subject to severe earthquake shaking yet. One of the few well-documented cases is the 150-m-high Zipingpu dam in China that was shaken on May 12, 2008 by a 7.9° Richter magnitude earthquake. Although experts reported that this CFR dam behaved remarkably well after the earthquake [²], other dam engineers reported the presence of significant face slab cracking, which was clearly appreciated because the reservoir was half of its maximum level when the earthquake occurred [³–⁵]. An enlightening lesson drawn from the seismic response of Zipingpu dam is that due to great stiffness differences between the rockfill and the concrete face, in-plane slab coupled with normal slab movements can cause large compression stresses that may exceed the compressive strength of the concrete, producing slab cracking and slab-joint waterstops damage, but no noticeable body dam damage. Accordingly, it is arguable that CFRDs are in the overall highly resistant to seismic loading since the concrete face and waterstops are damaged, the post-earthquake dam behavior will be hampered and major repairs should be performed before the dam operates properly again.

This paper presents a straightforward method developed from the results of nonlinear parametric analyses of 3D CFRDs to evaluate earthquake-induced in-plane transversal normal stresses in the concrete face of CFRDs. The final stress state developed by the construction and reservoir filling of each dam was assumed as the initial condition for the seismic computations. The authors are aware that a two-dimensional (upstream-downstream and cross river) orthogonal horizontal seismic environment would increase the concrete face in-plane motions and would mimic closer the actual seismic loading conditions [⁶]. Furthermore, all results included herein correspond to the resonance condition of all dams considered. It is important to point out that the earthquake-induced in-plane transversal normal stresses coupled with those caused by the reservoir filling increase the risk of waterstop rupture along the concrete face slabs vertical and peripheral joints [⁷]. Moreover, in extreme cases, concrete slab dislocations might occur, particularly at the upper central part of the concrete face when the reservoir is half-filled [⁸].

These straightforward procedures to assessing the behavior of the concrete slab after dam reservoir filling and seismic dam response were developed from a huge database generated using nonlinear static and dynamic three-dimensional numerical analyses on the same sample of CFRD’s, which was interpreted with the Buckingham Pi theorem [⁹] followed by the development of suitable relationships. Herein, the seismic procedure is first developed and then the full method of analysis (3D reservoir filling plus horizontal orthogonal 2D seismic loading) is applied to a hypothetical dam-case whose characteristics and input motion differ from those of the cases used for assembling the database.

The in-plane transversal normal stresses computed with the proposed method agree reasonably well with those acquired from the 3D finite difference computations. Accordingly, it can be argued that the simple method advanced in this paper can be used to compute the compression transversal stresses in the concrete slabs and consequently in the waterstops for their preliminary designs. This provides the dam engineer an innovative easy-to-use numerical tool to make reliable concrete face pre-designs.

Seismic numerical modeling of CFRDs

Dams models description

For paper completeness, Fig. 1 shows the overall geometric features of the CFRDs considered in the seismic analyses. As previously mentioned, the dam models were identical to those used in the simulation analyses of the construction and first dam reservoir filling [¹]. The dynamic parameters account the after-the-first-reservoir-filling stress-strain conditions to mimicking a continuous process. The dam-valley geometric parameters, the effect of which was analyzed, were maximum dam height (H), riverbed width (b), canyon slopes (T_v), and slab widths (S_j). The dimensionless factor (b/H) was considered equal to 0.5 throughout all the analyses. This value was chosen on the basis that the average of (b/H) obtained from 14 CFRDs built around the world was approximately 0.54 [¹].

The canyon slopes (T_v) were considered equal to take advantage of the symmetry in benefit of more refined finite element meshes; ergo, more reliable results. Considering the general CFRDs design practice, upstream and downstream slopes of 1.4H: 1.0V, crest width of 10.0 m, and a face slab thickness of 0.5 m were contemplated in this study. All 3D analyses were performed with the finite difference three-dimensional code FLAC^3D [¹⁰].

Modeling of constructive joints

Interface elements between the concrete slabs (vertical joints), between the slabs and the plinth (peripheral joints) and between the slabs and rockfill (transition joints) were employed in this study. These interface elements are similar to those proposed in the distinct elements method [¹¹] and coded in the FLAC^3D computer program. It is important to note that the stress-strain relationship for the joint tangential direction is assumed to follow Coulomb’s criterion. The shear force, F_s, acting on the interface node is limited by:

(1)F_{s\max }=cA+F_n\hspace{0.17em}\tan \phi

where c and f are the cohesion and friction angle at the interface surface; A is the influence area for specific nodes and F_n is the corresponding normal force. If F_s is greater than or equal to F_smax then F_s = F_smax. If the tension that exists across the interface exceeds the tensile strength of the joint (T), then the interface is broken and the normal and shear stiffnesses are cancelled. Consequently, the normal and shear stresses are not transmitted through the joint. The interface element behavior used in the numerical analysis of CFRDs was verified by static and dynamic tests on a laboratory model, which consists of a constant-weight rigid block, instrumented with accelerometers to measure block accelerations and a LVDT to monitor block displacements relative to the fixed bottom block (Fig. 2). The blocks’ arrangement is set up on a 1D shaking table. The characteristics of experimental and model tests are described in more detail elsewhere [¹²–¹⁴]. Herein only comparisons between experimental and theoretical results for the dynamic loading condition are shown.

It is common to consider the sliding characteristics of concrete face slabs as those of a rigid block, the dynamics of which are nonlinear due to the non-conservative nature of the friction phenomenon that develops within the vertical, peripheral and transition interfaces. Most analytical studies resort to Coulomb´s friction law, which implicitly means that a constant friction coefficient is being assumed. In this paper, the nonlinear friction law (Eq. (3)) proposed by Méndez et al. [¹³,¹⁴] is used to consider the time-friction variation of the interfaces. From equilibrium of block’s forces, we obtain the following expression:

(2)m{\ddot{U}}_g\cos \theta -m\left(\ddot{U}\cos \theta +g\sin \theta \right)=N\mu

where m is the block mass; Ü_g and Ü are the gravity acceleration and the input motion, respectively; m is the instantaneous dynamic friction coefficient and N is the varying normal force given by N=m\left(g\cos \theta +\left|{\ddot{U}}_g\right|\sin \theta \mathrm{sgn}{\ddot{U}}_g\right) where sgn is the signum function. To consider the friction force change, the static term is added to Eq. (2) to obtain the continuous friction coefficient variation (m) at any time

(3)\mu (t)=({\mu }_s-\tan \theta )-{\displaystyle \frac{\left(\left|{\ddot{U}}_g(t)\right|\cos \theta -\left|\ddot{U}(t)\right|\right)\cos \theta +g\sin \theta }{M{m}^{-\mathrm{1}}}}

For the sake of completeness, some of the model characteristics and test results are included. The fixed bottom block was 0.25 m long, 0.07 m wide and 0.07 m high, and the top block (weighing 1.557 kg) was 0.12 m x 0.07 m x 0.07 m, length, width and height respectively (see Fig. 3(a)). The static friction coefficient for the concrete-concrete interface, determined from previous studies, was 0.65 (f_s=33.2°). The kinetic friction coefficient was determined from shaking table tests applying a sinusoidal excitation. It is worth mentioning that the environmental conditions were held constant throughout the experimental investigations. These tests were modeled numerically with a three-dimensional finite difference-computing program using interface elements similar to those described in this section. The top and bottom blocks were modeled with solid elements (Fig. 3(a)) reproducing the physical and geometric characteristics of the laboratory model. Figure 3(b) compares the analytical and experimental top block-acceleration response time-histories, as well as the shaking table excitation. The interface was characterized by the time-dependent kinetic friction coefficient (Eq. 3), and normal and shear stiffnesses: k_n = k_s = 1.00E+08 kN/m². As indicated in Figure 3(b), the numerical model accurately reproduces the experimental results; consequently, it is deemed that the interface element with time-varying friction is suitable to model the behavior of slab-slab, slab-plinth and slab-rockfill contacts. The authors are aware that quantitatively the friction coefficient of the slab-rockfill interface will more likely be different, but the overall cyclic behavior will be alike. One of the objectives of an ongoing research is to shed additional light on this issue.

Dynamic properties of the CFR dam

The initial stress-strain conditions for all dynamical analyses were those corresponding to at the end of first dam reservoir filling. Details of their computations are presented in Sarmiento and Romo [¹]. These authors include some field investigations aimed at monitoring the clusters of extensometers and pressure cells located at several elevations and locations in three rockfill dams built in Mexico. The monitoring results clearly show that the behavior of the rockfill is practically linear up to the end of dam construction and remains nearly linear (having a slightly lower stiffness) upon reservoir filling completion [¹⁵,¹]. These measurements and confined laboratory testing results showed that rockfill Young’s modulus (E) is stress dependent according to a power law [E=E₀ (s_oct /s_{oct i}) ^a] where s_oct and s_{oct i} are the octahedral stresses (subscript i stands for initial), and a is a material fitting parameter [¹]. The volumetric weight, g_c, of the concrete and its Young’s Modulus, E_c, used in all the parametric analyses are given in Table 1. In addition, the Poisson’s ratio values of the rockfill and the concrete included in Table 1 were kept constant thru the analyses.

Evaluation of the dynamic properties of large particles assemblies such as the boulders used in rockfill dams is a difficult task mainly because of particle sizes. For example, to perform a cyclic triaxial test on a sample formed with, say, 20 cm size particles, and fulfilling the requirements of aligning at least eight particles along the diameter of the sample and of having a minimum slender ratio of 2.5, the cylindrical sample would be 4.0 m high. To perform dynamic tests on a sample 1.6 m in diameter and 4.0 m in height, laboratories would require colossal installations. A number of researchers have overcome this problem by scaling down the particle’s size and developed constitutive models (i.e., Refs [⁵,¹⁶–²⁰].). Their great contributions to the knowledge of rockfill dynamic behavior is beyond any doubt, but still there are unanswered questions as to how the scaling factor would affect the tests results; ergo, the constitutive models.

An alternate procedure to estimate in situ dynamic properties that has been in use for a number of years and proven to be reliable in forecasting the dam response to diverse earthquakes is to set up a proper seismic instrumentation to record input motions and the corresponding dam outputs at several points. This information is then used to solve the inverse problem and hence to estimate the in situ rockfill dynamic properties for small and large earthquakes.

In Mexico, several rockfill zoned and concrete face rockfill dams have been seismically instrumented, and the corresponding information collected. Unfortunately, in many cases, the dams’ inputs and the corresponding outputs were not simultaneously recorded, or the characteristics of the seismic events were rather similar. Nevertheless, the duality (input-output) phenomenon was properly recorded in a rockfill- central clayey core and we used this case to estimate rockfill in situ dynamic properties. The duality phenomenon was captured several times at El Infiernillo dam, which was seismically instrumented heavily (see Fig. 4) and various earthquakes having different characteristics were recorded allowing to defining the duality phenomenon. This fact offered the unique opportunity of solving the inverse problem for an appreciable number of excitation conditions that induced dam-cyclic deformations of different magnitudes and hence a range of shear strains, shear modulus and damping ratios. Using the obtained results, curves of normalized shear moduli and damping ratio versus developed dynamic shear strains were defined for the dam rockfill.

The procedures followed are described in more detail elsewhere [²¹–²⁴]. It is worth to mention that the theoretical model for the rockfill seismic behavior of El Infiernillo dam was developed based on a least square minimization approach followed by a trial and error procedure to tune up the dynamic behavior of all the materials integrating the zoned dam to improve the matching between computed and measured responses. Figure 5 compares the seismic response of the El Infiernillo dam shaken by a 5.5 Richter magnitude earthquake, with that calculated with the finite element method including the dynamic behavior model developed (Eqs. (4) and (5)). Comparisons between calculated with the measured responses at the crest of the dam (seismic station E, elevation 180) and point H within the embankment body (vertical array), due to the May 31, 1990 earthquake are shown in Figs. 5(a) and 5(b), respectively. It is worth stressing the fact that this earthquake occurred about a year after the model was developed (1988) and obviously, it was not in the model-development database. The results included in Fig. 5 show that the proposed model is capable of reproducing the recorded motions with a high degree of accuracy. Not only the amplitudes but the details of the measured responses are nicely reproduced. Accordingly, the comparison can be considered as a “prediction” type C, according to Lambe’s classification [²⁵].

When the optimum matching between calculated and measured responses to all seismic events was achieved, the corresponding set of properties was considered as the in situ dynamic characteristics of the various dam materials. This approach has been used in several previous rockfill dam studies leading to plausible results (i.e., [²⁶–²⁸]).

The model proposed as representative of the equivalent dynamic properties of compacted dam rockfill materials is given by the following hyperbolic relations [²²–²⁴],

(4)G=G_{\max }\left[\mathrm{1}-{\displaystyle \frac{(\gamma /{\gamma }_r)}{a+b(\gamma /{\gamma }_r)}}\right]

(5)\lambda ={\lambda }_{\min }+{\displaystyle \frac{(\gamma /{\gamma }_r)}{\left({\displaystyle \frac{\mathrm{1}}{{\lambda }_{\max }}}\right)+{\displaystyle \frac{\mathrm{1}}{({\lambda }_{\max }-{\lambda }_{\min })}}\left({\displaystyle \frac{\gamma }{{\gamma }_r}}\right)}}

where G_max is the maximum shear modulus (i.e., the modulus at shear strains around 10^-4%); l_min is the minimum damping ratio (i.e., value at shear strain values of 10^-4% ); l_max is the maximum damping ratio (i.e., value at a shear strain value in the neighborhood of 10%); and g_r, a, b are soil parameters given in Table 2 [²⁴].

The value of G_max was evaluated based on Seed and Idriss [²⁹] recommendation for granular materials,

(6)G_{\max }=\mathrm{1000}\hspace{0.17em}{K}_{\mathrm{2}}{({\sigma }_m)}^{\mathrm{0.5}}

where s_m is the mean normal effective stress in lb/ft² and K₂ is a soil parameter that depends mainly on the void ratio.

Seismic input

Figure 6 depicts the input motion used in the seismic analyses carried out in this research. This harmonic function considers a peak ground acceleration of 0.3 g, a duration of 10 s, the frequency of which is set equal to the natural frequency of each dam model considered in this study. This signal is applied synchronously in two orthogonal horizontal directions (upstream-downstream and cross river) at the base of the finite difference model as indicated in Fig. 7. The plinth was assumed to be attached to the rock material dam foundation, which was considered much stiffer than the dam material. As mentioned, an interface element was included between the plinth and the concrete face slabs.

Although the dam models used in these analyses are homogeneous and sited on regular and symmetrical canyons, most key aspects of the phenomena under study are considered. Figure 7 shows an example of the 3D numerical models used in this study for H=60 m, b=30 m, and T_v=1. The maximum size of the elements fulfills the criterion proposed in previous investigations [³⁰,³¹]: it should be one fifth of the minimum shear wave velocity in the dam divided by the maximum frequency (8.0 Hz) considered in the analyses.

In-plane concrete face stresses at the end of reservoir filling

Sarmiento and Romo [¹] proposed Eqs. (7) and (8) to estimate the maximum in-plane compressive and tensile stresses that develop in the concrete face upon reservoir filling:

(7)$$\begin{gathered} {\sigma }_{c\left(\max \right)}={\sigma }_{cjo\mathit{int}less}A{(L/H)}^{-B} \\ A=0.020S_j+0.19545B=-0.0098S_j+0.346 \end{gathered}$$

(8){\sigma }_{cjo\mathit{int}less}=\frac{{\gamma }_c\sqrt{A_L}}{\alpha }[-B{\exp }^{-0.227(L/H)}]\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}B=0.3431{\alpha }^{0.4813}

where s_{c jointless} are maximum in-plane compressive stresses developed in a jointless concrete slab upon reservoir filling, calculated with Eq. (8); A_L is the concrete face area, and a, L and H have been previously defined.

Concrete Face Seismic Response

Dynamic in-plane normal stresses in the concrete panels

Figure 8 shows the variation of in-plane transversal normal stresses, s, in concrete slabs for the conditions: at the end of reservoir filling (contours) and at the end of reservoir filling plus seismic load (selected points) for the model with H = 60 m, b = 30 m, S_j = 10 m, T_v = 1.0, and a = 0.40. Similar cyclic stress trends were obtained for other rockfill dam-concrete face characteristics considered to develop the database [³²]. Due to paper length restrictions, the corresponding results are not included herein, but the analytical expressions given bellow reflect their effects.

The stress-time plots in Fig. 8 include the response when the seismic excitation is applied only in the upstream-downstream direction (X), and also along the two horizontal orthogonal directions (XY) at selected points (P-1 to P-8). The graphs in this figure clearly show that the stress-time responses vary spatially and that the consideration of both horizontal components as input affects appreciably the magnitude of the concrete face in-plane dynamic stresses. Is important to note that the intensity of the dynamic stresses increases as we draw closer to the abutments. Notice that all pictures in this figure have a vertical shift at their origin (time equal zero); the initial values correspond to the after-first reservoir filling-induced stresses. Accordingly, the stresses shown include the full loading history. It is noticeable that compressive and tensile stresses reach much higher magnitudes, particularly in the upper third of the lateral slab L-6 (point P-8) than those generated in the central slab L-1 at the same elevation (point P-4). These earthquake-induced in-plane transversal normal stresses increase the risk of concrete slab cracking and negatively influence the behavior of the construction joints, augmenting the possibility of their rupture or dislocation, mainly along the upper third of the vertical and peripheral joints. At this stage, it is important to bring the reader’s attention to the importance of considering the seismic environment composed by along-river coupled with cross-river motions. Figure 8 clearly shows the significant effect this has on the in-plane concrete slab stress responses as compared with those obtained when only the upstream-downstream component is considered. This is particularly more significant as we move toward the valley slopes. It is important to stress the fact that in most dam designs only the upstream-downstream seismic component is considered.

Effect of spacing between vertical joints

Among the many factors that affect the seismic response of the concrete face, one of great influence is its flexibility mainly influenced by the number of concrete slabs that integrate it. Accordingly, the spacing between vertical joints (S_j) is a parameter the effect of which regarding the seismic response of the concrete face of CFRDs, should be assessed particularly for the in-plane transversal normal stresses. Figure 9 shows the variation of the normalized in-plane maximum dynamic stresses throughout half of the width of the concrete face at three elevations (z/H= 0.4, 0.6 and 0.8). It is important to mention that the normalized stresses given in Fig. 9 and thereafter are the absolute values of the earthquake-induced maximum in-plane dynamic stresses at specific elevations. In the ordinates, the maximum seismic stresses are normalized with respect to the maximum seismic absolute stress developed at the central concrete slab (y= 0 m) and the corresponding elevation. In the abscissas, the-y- horizontal distance is normalized with respect to L_z which is the gorge width at the-z- elevation (see inset in the third plot of Fig. 9).

Notice that the maximum normalized seismic stresses for all cases studied here occur at a distance y/L_z = 0.45 from the center of the valley (y=0), and they develop at elevation z/H=0.4. As to the influence of S_j on in-plane transversal normal stresses, the results indicate that as the number of vertical joints increases the in-plane dynamic stresses in the concrete slabs attenuate more, indicating that it would be beneficial to build narrower concrete slabs. However, the construction procedure becomes more cumbersome as S_j decreases. Hence, according to the results presented in Fig. 9, it seems that S_j = 10 m would be an adequate practical concrete slab width to integrate the concrete face whenever possible. So from here onwards the analyses consider only a value of S_j equal to 10 m. Eq. (9) is the best fit to the results for T_v =1.0 shown in Fig. 9:

(9)\hspace{0.17em}{\sigma }_{seismic}={\sigma }_{seismic\hspace{0.17em}(y=\mathrm{0})}[A\hspace{0.17em}\exp {\hspace{0.17em}}^{B(y/L_z)}]

Where the parameters A and B, are given in Table 3. The coefficients of determination (R²) for Eq. (9) vary from 0.872 to 0.927 for the dam cases considered in this study; the R² intervals are indicated in each of the three plot’s set that integrates Fig. 9. Note that the coefficient of determination (R²) decreases slightly as z/H increases from 0.4 to 0.8 for any of the cases included in this research.

Effect of valley slopes

Figure 10 shows the normalized in-plane maximum stresses induced at three elevations in the concrete face (z/H= 0.4, 0.6 and 0.8) by dynamic loading on three CFRDs models. Dam height, riverbed width and the spacing between vertical joints were kept constants (H=120 m, b=60 m, S_j=10 m), and valley slopes considered were T_v=1.0, 1.5 and 3.0. The results included in Fig. 10 show that the maximum values of the normalized stresses follow a bit different trend, which depends on the dam slope gorge. While for T_v =1 the seismic stress ratio increases from the dam central section to the dam abutment regardless of the dam height, for higher canyon valley slopes the stress ratio reaches a maximum and then starts decreasing. Notice that as we approach the dam crest the stress ratio amplification is reduced as the valley slopes become steeper and the distance y/L_z from the dam center (y=0) to the peak of normalized stress maxima curves becomes longer.

In order to develop a general expression linking (y/L_z) and the stress ratio, the trends followed by the curves in Fig. 10 were simplified as shown in Fig. 11(a). Notice that the stress ratios were kept constant once they reached their maximum value. The best fitting expression to this set of curves is given by Eq. (10). The coefficients of determination (R²) of the Eq. (10) varies from 0.841 to 0.943, which are included in each graph that integrates Fig. 11. It is worth noticing that with this simplification, all trends observed in this paper for abutment slopes from 1.0 to 3.0 are alike.

(10)

Effect of dam height

Figure 12 shows the earthquake-induced maximum normal transversal dynamic stresses across the concrete face at three elevations: z/H=0.4, 0.6 and 0.8. Stress normalization was carried out with respect to the maximum stress developed at same elevation in the central section. It is noteworthy that dam heights have a rather small influence on the normalized transversal stresses in the concrete face. The coefficients of determination (R²) of the fitted trend curves through the points (finite difference computed values) are included in Fig. 12, which vary from 0.94 to 0.98.

In-plane dynamic stresses in the concrete face mid-section

Figure 13 shows the profile of maximum in-plane transversal normal dynamic stresses in the concrete central slab (y=0), normalized with respect to maximum in-plane normal stresses induced by reservoir filling at same elevations along the central slab. These results yield a similar trend (solid line) for any dam height, slab widths and canyon slopes, regardless of the static stiffness of the rockfill (power law exponent a). However, as the exponent a increases the normalized dynamic stresses increase a bit for equal normalized dam elevation.

The analytical representation of all data included in Fig. 13 is given by Eq. (11). The coefficients of determination (R²) vary from 0.975 to 0.986.

(11)\begin{array}{l}{\sigma }_{\hspace{0.17em}seismic\hspace{0.17em}(y=\mathrm{0})}={\sigma }_{c\hspace{0.17em}filling\hspace{0.17em}(y=\mathrm{0})}[F\hspace{0.17em}{\exp }^{G\hspace{0.17em}(z/H)}]\\ F=-\mathrm{0.0096}{\alpha }^{\mathrm{2}}+\mathrm{0.0280}\alpha -\mathrm{0.0072}\\ G=\mathrm{5.6891}{\alpha }^{\mathrm{2}}-\mathrm{8.5067}\alpha +\mathrm{10.8029}\end{array}

Where, in-plane transversal normal stresses upon first reservoir filling at the concrete face central section (s_{c filling (y= 0)}) to any elevation (z/H), can be calculated with Eq. (12) proposed by Sarmiento and Romo [¹].

(12){\sigma }_{c\hspace{0.17em}\hspace{0.17em}filling\hspace{0.17em}(y=\mathrm{0})}={\sigma }_c{\hspace{0.17em}}_{(\max )\hspace{0.17em}}[\mathrm{6.165}{(z/H)}^{\mathrm{3}}-\mathrm{13.041}{(z/H)}^{\mathrm{2}}+\mathrm{7.115}(z/H)-\mathrm{0.185}]

where s_{c (max)} is the maximum in-plane compression induced by reservoir filling and can be calculated using Eqs. (7) and (8).

It is important to note that the earthquake-induced stresses along the mid-section of the concrete face (see Fig. 13), are smaller from the dam’s base up to 60% of their height than those caused by the first dam reservoir filling. Above dam height z/H=0.6, the normal seismic stresses become bigger than those induced by the first filling and their ratio s_seismic/s_filling increases rapidly. Accordingly, first reservoir filling and earthquake-induced in-plane transversal normal stresses should be defined as accurately as possible to properly design waterstops to avoid water leakages that could turn dam’s operation inefficient or even worse be the feeble links that trigger localized failures on the concrete face.

It seems appropriate, at this point, to emphasize that when designing the concrete slabs and the corresponding waterstops, the total in-plane transversal normal stresses developed should be considered. That is to say, the added effect of the first-reservoir-filling and the earthquake-induced in-plane normal stresses as indicated in this paper.

It is important to stress the fact that R² describes how well any model developed from a database performs. As shown in the above presented results, all R² values are bigger than 0.84 and as it is well known that as long as the R² is larger than 0.8 the model developed from a particular database will accurately forecast the behavior of structures equivalent to those included in the database as is shown in chapter 5.

However, it is worthwhile to mention that the model advanced could be upgraded by stochastic modelling following the procedure proposed elsewhere [³³–³⁷].

Application example

To show how the simple method of analysis can be used, the following hypothetical CFRDs is used: H=140m, b=70m, T_v=1.0, S_j=10 m, and a=0.60. The results obtained with the expressions advanced in this paper are compared with those computed with the 3D numerical model shown in Fig. 14(a). The signal shown in Fig. 14(b), defined from the recorded earthquake on rock at the right abutment of El Infiernillo dam on September 19, 1985 in Mexico, was used as (upstream-downstream coupled with cross-river) input motion. The maximum input acceleration was 0.3g, the predominant frequency of the excitation of the acceleration time history coincides with the fundamental frequency of the CFR dam (1.95 Hz). This seismic excitation was applied at the rigid boundary of the dam, considering two equally intense orthogonal horizontal components: upstream-downstream and the dam axis directions.

To calculate the in-plane transversal normal stresses with the simplified method advanced in this paper follow the next process:

a. With Eqs. (7) and (8) maximum in-plane compressive stresses in the concrete face at the end of the reservoir filling are calculated.

b. Compute the in-plane transversal normal stresses profile in the central concrete slab developed at the end of the reservoir filling using Eq. (12).

c. With the information obtained in the previous paragraph and Eq. (11), the earthquake-induced maximum in-plane transversal normal stresses in the central concrete slab are calculated. From the comparison between the results obtained with both procedures shown in Fig. 15, it can be concluded that both procedures yield similar results (coefficient of determination R² is equal to 0.901).

d. Finally, in Fig. 16 the maximum in-plane transversal normal stresses induced by earthquake loading at three elevations in the concrete face obtained with the procedure proposed, Eqs. (9) and (11), are compared with those evaluated with the 3D numerical method. The results compare very well. The coefficients of determination (R²) vary from 0.96 to 0.99.

Conclusions

In the design of concrete slabs in CFRDs the stress state induced in the concrete face by the first-dam reservoir-filling plus earthquake loading must be known. Author’s awareness of the difficulties involved in setting up the numerical models to carry out the analyses required to compute properly the concrete face in-plane transversal normal stress states, in this paper a simplified procedure to estimate such stress states is advanced. This procedure was developed from 3D analyses, the results of which were digested and interpreted using dimensionless theory. The proposed method requires only the sequential use of a number of expressions included in the paper. The example worked in the article indicates how the procedure can be used to estimate the normal in-plane stresses in the concrete face. It is worth mentioning that the geometrical characteristics of the CFRD and the input dynamic motion used in the example were different from those used in the parametric investigation. Accordingly, the results obtained with the advanced simple procedure are reliable and henceforth useful for preliminary designs of actual CFRDs.

According to the results presented in this study, the dynamic in-plane transversal normal stresses increase across the concrete face. These results indicate that special emphasis should be placed on concrete slabs (and waterstops) mainly at their upper third, not only in the central zone of the concrete face, but also in lateral slabs where the in-plane transversal normal stresses increase notably. The above condition is most unfavorable in higher dams, narrower valleys, and when the width of the concrete panels increases.

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