In-plane transversal normal stresses in the concrete face of CFRD induced by the first-dam reservoir filling

Abstract

To evaluate the effects of dam height, valley narrowness and width of concrete slabs on the first-dam reservoir filling in-plane transversal normal stresses in the concrete face of CFRD´s, 3D finite difference analyses were carried out. Behavior of rockfill dams considered in this study was defined from the monitoring of a number of 3D sets of pressure cells and extensometers installed in three large dams in Mexico. The 3D analyses results show that high in-plane transversal compressive stresses develop within the concrete panels located in the central concrete face zone upon dam reservoir filling loading. Likewise, in-plane induced tensile transversal stresses in the zones near the abutments increase the potential of slabs cracking and damaging the waterstops in-between the vertical and perimetral joints. From the results of the 3D finite difference analyses, a simple method to estimate in-plane normal stresses in the concrete face is advanced and through comparisons with the results of a 3D case numerical study, its accuracy assessed.

Keywords

concrete face / CFR dams / reservoir filling / slab in-plane stresses

Introduction

The concrete face of the CFRDs endure large distortions upon their reservoir filling, the magnitude of which depends greatly on the compressibility of the rockfill that supports the concrete face. Indeed, recorded normal concrete slab displacements of a large number of CFRDs upon hydraulic loading reach values about 0.1% of dam height, located at heights ranging between 30 to 50% of the maximum height of the dam [¹–³]. In addition, case histories have shown that concrete slabs move toward the center of the canyon causing compression stresses in the central zones of the concrete face and tension stresses in areas closer to the abutments and near the dam crest. This tendency will be more acute as the slopes of the gorge walls are steeper. Accordingly, while concrete slab vertical joints in the central areas of the concrete face will close, the perimetral joint and those near the abutments will tend to open.

A number of CFRDs have endured severe concrete face cracking during the first reservoir filling (i.e., the 202-m-high Campos Novos dam in Brazil [⁴], the 185-m-high Barra Grande dam in Brazil [⁵], the 145-m-high Mohale dam in South Africa [⁶], and the 178-m-high TSQ-1 dam in China [⁷]). These dams presented ruptures along central vertical joints, and some gaps between the face slab and the rockfill [⁸]. Although there is a number of factors that possibly caused different damages in these dam concrete faces, some authors believe that they were due to higher transversal and longitudinal compressive stresses that developed at the central region of the concrete face (⁵,⁹). Generation of these large stresses is likely due to a rather poor compaction of the rockfill located at the upstream side of the dam axis. Accordingly, one can reduce or even eliminate most of the above-mentioned damages that the concrete slabs suffer upon reservoir filling by ensuring a highly compacted rockfill placed at the upstream portion of the dam where stresses induced by dam reservoir filling are higher. With this in mind, the design of El Cajon and La Yesca dams in Mexico included essentially three zones: the upstream-zone heavily compacted, the central transition-zone compacted with lower energy, and the downstream zone compacted with the least energy. The boundary contacts between the transition zone and the up- and downstream zones are 0.5H:1.0V sloping outwards. Their behavior, after a number of years following their construction and first reservoir filling has been very good (i.e., [¹⁰–¹²]). These case histories and the results included in this paper, lead to the conclusion that the key recipe to ensure a proper behavior of the dam concrete face is that the compaction of the rockfill material located upstream be of high quality so that it is placed at a void ratio near the minimum void ratio of the rockfill used. Hence, the upstream dam region be stiff enough to avoid concrete slabs developing upstream curvatures (mainly at the upper third of the dam) thus eliminating the risk of cracking and concrete slab joint opening. Moreover, CFRDs overall long-term behavior will be better. Furthermore, a highly compacted rockfill ensures that the CFRD behavior remain in the elastic range. Regarding the creep effects, on the properties of the concrete slab, are most likely negligible because the slabs are mainly supported by the rockfill; however for special cases the creep phenomenon influence on long-term behavior should be considered. In this paper, it was not accounted (creep and temperature effects) for given the relatively short time that takes to build a CFRD and fill its reservoir.

To evaluate the influences of dam height, valley narrowness (considering a number of valley slopes) and width of concrete slabs on the first-dam reservoir filling induced transversal in-plane stresses in concrete face of CFRD’s, parametric analyses of 3D CFRDs were carried out using the well-known computer program FLAC^3D [¹³]. Then, using the Buckingham’s Pi theorem, a procedure to assess the in-plane transversal stresses induced by first-dam reservoir filling loading, was developed. To assess its accuracy, the advanced method was applied to a hypothetical example, different from those included in the parametric analyses, and then compared with the results obtained from a 3D analysis of the same example. The in-plane transversal stresses computed with the proposed method agree reasonably well with those yielded by the 3D finite difference computations. Accordingly, the simple method advanced herein can be used to compute the transversal stresses in the concrete slabs and concurrently in the in-between waterstops for their preliminary designs. This provides the design dam engineer an innovative easy to use tool to make reliable early CFRD assessments.

Numerical modeling of CFRDs

Dams geometric characteristics

The parameters defining the overall geometry of the dams were its maximum height (H), the riverbed width (b), the upstream and downstream slopes (T_c), and the gorge wall slopes (T_v). The concrete face consisted of slab panels and joint waterstops as in actual CFRDs. Figure 1 shows the geometric features of the CFRDs considered in the parametric analyses. All 3D analyses were performed with the code FLAC^3D [¹³].

To assess the effect of the gorge narrowness, the dam height and slab panels width, a number of values of the parameters T_v (1.0, 1.5 and 3.0), H (60 m, 120 m and 180 m) and S_j (10 m, 20 m and 30 m) were assumed. Three riverbed widths (b) were considered (30 m, 60 m and 90 m), maintaining the dimensionless factor b/H equal to 0.5 throughout all the analyses. This value was chosen because the average of (b/H) obtained from 14 CFRDs built around the world was 0.54. Notice that this factor together with the valley slope, T_v, define the topography of the canyon. 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 considered in this study. The canyon slopes (T_v) were considered equal to take advantage of the symmetry in benefit of more refined finite element meshes, although the authors acknowledge the potential influence of asymmetric canyons. Accordingly, strictly speaking the method is valid for symmetric canyons, although it could be applied to ostensible asymmetric canyons considering the two slopes to obtain upper and lower limits of the in-plane stresses that would be helpful for preliminary designs.

Interface elements

Interface elements between the concrete slabs (vertical joints), between the slabs and the plinth (perimetral joint) and between the slabs and rockfill (transition joint) were included in the model, as indicated in Fig. 1. Interface elements allow the development of relative translational movements among concrete slabs, rockfill, and plinth. Accordingly, the actual physics of the problem is as closely as possible modeled and hence the computed stresses in the concrete face caused by hydrostatic loading are more reliable. Most interface elements commonly used in problems of soil-structure interaction (i.e., Refs. [¹⁴–¹⁷]) consider that the properties that define a discontinuity are normal stiffness, k_n, and shear stiffness, k_s, which relate the average normal and shear stresses to the respective joint displacements as follows:

(1)\begin{array}{l}{\sigma }_j=k_n{\delta }_n,\\ {\tau }_j=k_s{\delta }_s.\end{array}

The interface elements employed in this study are similar to those proposed in the distinct elements method [¹⁸] and coded in the FLAC^3D computer program: 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:

(2)F_{\mathrm{s}\max }=\hspace{0.17em}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 (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 concrete-concrete friction coefficient of the interface elements used in the numerical analysis of CFRD was obtained from static tests on a model, which consisted 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 (see Fig. 2(a)). The blocks’ arrangement is set up on a 1D shaking table. The characteristics of experimental and model tests are described in more detail elsewhere [¹⁹,²⁰].

The fixed bottom block was 0.25 m long, 0.07 m wide and 0.07 m high, and the sliding top block (weighing 1.557 kg) was 0.12 m × 0.07 m × 0.07 m, length, width and height respectively (Fig. 2(a)). From a large amount of tests that basically consisted on slowly inclining the pad, as shown in Fig. 2(a), the mean friction coefficient value was 0.53 (f_s = 28°). It is worth mentioning that the environmental conditions were held constant throughout the experimental investigations. To assess the reliability of the 3D finite difference-computing program used in the parametric analyses, the experiments were numerically mimicked with this software. The top and bottom blocks were modeled with solid elements (Fig. 2(b)) reproducing the physical and geometric characteristics of the laboratory model. The mean value of the slope of the pad computed when the top block started sliding was slightly smaller (27°). In the parametric studies, the concrete-concrete friction was considered 0.53, and the concrete-rockfill friction was assumed 37% larger to account for potential construction and concrete-rockfill roughness effects on the concrete-rockfill interface [²¹]. The values of the friction coefficients as well of the normal and shear stiffnesses are included in Table 1. It is worthwhile mentioning that care was taken to avoid that the nodal points corresponding to the concrete slab finite difference mesh did not overlap or cross the line that link those on the upstream rockfill dam slope, particularly on the lower third of the embankment where higher hydraulic pressures act on the concrete face. To achieve this, the concrete-rockfill interface normal stiffness (k_n) values were increased up to 100 MPa, as indicated in Table 1.

Material properties

The volumetric weight, g_c, of the concrete and its Young’s Modulus, E_c, assumed in all the parametric analyses are given in Table 2. In addition, the Poisson’s ratio values of the rockfill and the concrete included in Table 2 were kept constant thru the investigation.

Many laboratory and in situ tests and numerical computations point out to the fact that the rockfill Elasticity Modulus, E, is stress dependent. Such modulus-stress dependence was modeled in the analyses by the power Eq. (3). The modulus E₀ in Eq. (3) means the rockfill modulus for a normal octahedral stress confinement equal to 196 KPa.

Equation (3) was obtained from in situ plate test results carried out on trial rockfill embankments and laboratory tests performed for various rockfill materials included in a number of dams built in Mexico (i.e., El Cajon, La Yesca and La Parota dams [¹⁰,¹¹]). It is worth noting that with the incorporation of this power expression in the software used in the parametric analyses, the spatial variation of E is automatically considered. Hence, the many stress paths developed within the dam during its construction and first reservoir filling are accounted for.

(3)\mathit{E}={\mathit{E}}_{\mathrm{0}}{\left({\displaystyle \frac{{\sigma }_{oct}}{{\sigma }_{oct\hspace{0.17em}i}}}\right)}^{\alpha },

where s_oct is the octahedral stress within the dam for any construction stage including the reservoir filling; s_{oct i} corresponds to the minimum stress used (195 KPa) in laboratory tests carried out on the various rockfills used to build the above mentioned CFRDs; a is a setting parameter, which was considered equal to 0.4, 0.6 and 0.8 in the parametric analyses. The E_c of the slab and the plinth was considered linear elastic and its strength was 30 MPa.

From very thorough investigations, Alberro et al. [²²] concluded that the rockfill material within a dam behaves linearly. These investigations included monitoring the clusters of extensometers and of pressure cells installed in three rockfill Dams built in Mexico at several elevations and locations as indicated in Fig. 3(a) for the Aguamilpa CFRD. The results in terms of σ_oct− ε_oct and t_oct− g_oct obtained from two clusters, one of pressure cells and other of extensometers located at dam elevation 100.90 m (Cluster G-14), are shown in Figs. 3(b) and (c). They 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. Furthermore, from these two figures it can be concluded that the rockfill in Aguamilpa CFRD was well compacted due to the rather small octahedral strains developed under significant loading. Similar results were obtained for other clusters distributed throughout the Aguamilpa dam embankment and clusters installed within the rockfill of El Caracol and Peñitas dams for ε_oct up to 1.1%. Accordingly, it was deemed adequate to carry out the 3D analyses for the parametric study considering the spatial variation of E according to the octahedral stress state and consider the σ_oct– ε_oct relationship linear during numerical mimicking of dam construction and reservoir filling. It is important to point out that the potential effect of time was not considered explicitly. However, having in mind that to limit dam deformations upon construction and reservoir filling the rockfill must be highly compacted and furthermore rockfill particles must be sound, it is likely that dam embankment long-time deformations be smaller than those developed upon reservoir filling. Accordingly, the simple model advanced herein to estimate the in-plane slab stresses is applicable for preliminary concrete face designs. To account for the influence on in-plane stresses of possible long-term deformations it is advisable to increase by 15% the computed in-plane stress values.

The results obtained by Alberro et al. [²²] for the behavior of rockfill and also for the core material (Peñitas and La Villita dams) clearly indicate that the s_oct– ε_oct relationship is linear and the modulus E varies according to a potential law (Eq. (3)). This seems to contradict the general believe and what some investigations on rockfill materials have yielded [²³–²⁶] as to the nonlinear behavior of rockfill. This difference in opinions can be explained as follows: The above researchers performed tests on cylindrical samples submitted to all around confining pressure applied with a fluid (e.g., water). Although the field stress lateral boundary conditions can be met in the laboratory tests, the shear stiffness of the confining liquid is practically nil as compared with that of the material neighboring a virtual cylindrical sample in the dam. Accordingly, while the laboratory sample deforms laterally with essentially no restrain, the virtual sample in the field interacts with the neighboring material that has about the same stiffness providing further confinement, not modeled in the triaxial laboratory tests, that greatly constrains the lateral displacement of the rockfill. Consequently, the virtual field sample deforms mainly vertically as occurs in odometer-type tests. Therefore, when the rockfill is well compacted it is unlikely that large deformations and hence nonlinear behavior of the embankment dam does not develops.

CFRD construction modeling

The analyses of the CFRDs considered the following staged process. 1) Construction of the rockfill embankments was mimicked in ten stages; 2) the plinth and the concrete face were placed in two consecutive stages; and 3) a hydrostatic load on the concrete face of the dams was applied in four stages to simulate the reservoir filling at the maximum level. Figure 4 shows some of the 3D numerical models used in this study. The insets in this figure define the characteristics of each model. In these analyzes the dam body and the plinth were assumed fixed to the valley, which is considered rigid (all points along the dam-foundation contact are fixed), but the neighboring concrete slabs could have relative movement with respect to the plinth (slab-plinth interface).

In-plane concrete face stresses upon reservoir filling

Figure 5 shows contours of the transversal compression-tension stresses (s_c, s_t) across the concrete face for the at-the-end of reservoir filling, for the case of 60.0 m dam height, 30.0 m riverbed width, valley slopes 1.0H: 1.0V, and spacing between vertical joints, S_j of 10.0 m. It is worth mentioning that similar contour distributions were obtained for other dam’s heights, riverbed widths and vertical joint separations [²⁷]. Due to space restrictions, they are not included in this paper, but the interested reader can consult the above-mentioned reference. The maximum in-plane compression stresses developed in the central concrete slab at an approximate elevation of z/H = 0.4. On the other hand, the maximum in-plane tensile stresses occur near the valley slopes, at elevation z/H = 0.3. These results follow the displacement patterns referred to previously: while the perimetral and the neighboring joints will tend to open upon dam reservoir filling, the vertical joints in the central areas will undergo compressions. Accordingly, the design of the water seals should consider the stress spatial variation likely to develop within the concrete face to ensure their water tightness. These results agree qualitatively with the concrete slab displacements observed in the field and reported by a number of authors [1,2]. The inset in Fig. 5 represents the general distribution of the in-plane stresses in the concrete face of CFRDs. In the next paragraphs, we propose simple expressions to evaluate the maximum stresses induced by reservoir filling loading.

Maximum in-plane stresses

Influence of concrete slab width

Maximum in-plane compressive and tensile stresses, normalized with respect to the maximum in-plane stresses developed in a jointless concrete slab upon reservoir filling are shown in Fig. 6, for a = 0.8. It is important to stress the fact that the results are very similar for other values of a. The results shown in this figure, for the CFRDs cases considered in the study, depict a clear tendency indicated by the solid lines. Maximum in-plane compressive and tensile stresses increase with slab width. It is important to point out that for the analytic expression’s development, absolute values of the stresses were considered. Figures 6(a) and (b) show the influence of the concrete slab width on the compression and tension normalized stress magnitudes. For S_j = 10 m, the compressive stress ratio (with joints / jointless) is about 0.29 (Fig. 6(a)) and remains almost constant for any canyon section and dam height considered in this study. For S_j = 30 m, a stress ratio around 0.75 (Fig. 6(a)) is obtained and keeps on practically constant for all CFRDs analyzed in this research. These results seem to indicate that as the concrete slab width increases the stress ratio tends to one, which means that the concrete face is continuous (without joints) as would be expected. Furthermore, these results indicate that the in-plane normalized compression stresses are practically independent of the b/H ratio. On the other hand, the results for the maximum in-plane tensile stresses in Fig. 6(b) clearly show the significant valley geometry effect (L/H). Notice that the stress ratio decreases as the valley becomes wider. Stress ratios vary between 0.10 (L/H = 7.0) and 0.25 (L/H = 2.0) for S_j = 10 m, and between 0.20 (L/H = 7.0) and 0.30 (L/H = 2.0) for S_j = 30 m. The results obtained for the CFRDs cases considered in this study yield a clear tendency as pointed out by the solid lines.

These results will certainly provide guidance in taking decisions during the preliminary design stages of CFRDs. Equations (4) and (5) give the expressions that best fit these curves:

(4)\begin{array}{c}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}:\\ \square \end{array}\begin{array}{c}\hfill {\sigma }_{\mathrm{c}}={\sigma }_{\mathrm{c}}{}_{\hspace{0.17em}\mathrm{j}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{s}}\hspace{0.17em}\mathit{A}\hspace{0.17em}{\left(\mathit{L}/\mathit{H}\right)}^{-\mathit{B}}\hfill \\ \mathit{A}=\mathrm{0.020}\hspace{0.17em}{\mathit{S}}_{\mathrm{j}}+\mathrm{0.19545}\mathit{B}=-\mathrm{0.0098}\hspace{0.17em}{\mathit{S}}_{\mathrm{j}}+\mathrm{0.346.}\hfill \end{array}

(5)\begin{array}{c}\mathrm{T}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}:\\ \square \end{array}\begin{array}{l}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\sigma }_{\mathrm{t}}={\sigma }_{\mathrm{t}}{}_{\mathrm{jointless}}\hspace{0.17em}\mathit{C}\hspace{0.17em}{\left(\mathit{L}/\mathit{H}\right)}^{-\mathit{D}}\hfill \\ \mathit{C}=-\mathrm{0.0022}\hspace{0.17em}{\mathit{S}}_{\mathit{j}}+\mathrm{0.4183}\mathit{D}=-\mathrm{0.0238}\hspace{0.17em}{\mathit{S}}_{\mathit{j}}+\mathrm{0.9785.}\hfill \end{array}

Effect of valley narrowness and dam height

The maximum in-plane compressive and tensile stresses in a jointless concrete face induced by dam reservoir filling included in the above equations, normalized with respect to the dimensionless factor γ_cA_L^0.5 (where γ_c = 24 kN/m³ is the volumetric weight of concrete, and A_L is the concrete face area) are plotted in Fig. 7. It is worth mentioning that the concrete face area A_L contains important information of the valley geometry, and its square root yields an equivalent width (or height) of the concrete face [³,⁵,⁸].

The valley narrowness and dam height effects are appreciated in the plots included in Fig. 7, where the compressive and tensile stresses increase when the ratio (L/H) is smaller and the dam height increases. The solid lines are the trends of the reported values for each dam height, and Eqs. (6) and (7) give the functions that best fit these results, which yield the means to compute the maxima in-plane compressive and tensile stresses in a jointless concrete face:

(6)\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}:{\sigma }_c{}_{\mathrm{jointless}}\hspace{0.17em}={\displaystyle \frac{{\gamma }_c\hspace{0.17em}\sqrt{{\mathit{A}}_{\mathrm{L}}}\hspace{0.17em}}{\alpha }}[-B\hspace{0.17em}{\exp }^{-\mathrm{0.227}(L/H)}]\hspace{1em}where\hspace{1em}\mathrm{B}=\mathrm{0.3431}\hspace{0.17em}\alpha {\hspace{0.17em}}^{\mathrm{0.4813}}\hspace{1em},

(7)\mathrm{T}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}:{\sigma }_t{}_{\mathrm{jointless}}\hspace{0.17em}={\displaystyle \frac{{\gamma }_c\hspace{0.17em}\sqrt{A_L}}{\alpha }}\hspace{0.17em}[C\hspace{0.17em}{\exp }^{-D\hspace{0.17em}(L/H)}],

where the parameters C and D are given in Table 3.

In-plane stresses at the central section of the concrete face

Figure 8 shows the normalized in-plane stresses upon first reservoir filling at the concrete face central section (y = 0 m) to any elevation (z/H). The in-plane stresses induced at any depth upon dam reservoir first filling were normalized with respect to maximum in-plane compression generated along the central section of the concrete face. Equation (8) analytically expresses this normalized stress vertical variation:

(8){\sigma }_{\mathrm{c}\hspace{0.17em}\hspace{0.17em}\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{g}\hspace{0.17em}(\mathit{y}=\mathrm{0})}={\sigma }_{\mathrm{c}}{\hspace{0.17em}}_{\left(\max \right)\hspace{0.17em}}\left[\mathrm{6.165}{\left(\mathit{z}/\mathit{H}\right)}^{\mathrm{3}}-\mathrm{13.041}{\left(\mathit{z}/\mathit{H}\right)}^{\mathrm{2}}+\mathrm{7.115}\left(\mathit{z}/\mathit{H}\right)-\mathrm{0.185}\right],

where s_{c (max)} is the maximum in-plane compression induced by reservoir filling and may be calculated using Eqs. (4) and (6).

Application example

As an example to show how the simple method of analysis can be used, the following hypothetical CFRD is used: H = 140 m, b = 70 m, T_v = 1.0, S_j = 10 m and a = 0.6. The results obtained with the expressions advanced in this paper are compared with those computed with the 3D numerical model shown in Fig. 9.

To calculate the in-plane normal stresses with the simplified method advanced in this paper the following procedure should be followed:

1) With Eqs. (4) and (5) maximum in-plane compressive and tensile stresses in the concrete face at-the-end of the reservoir filling are calculated. In Table 4, the values calculated with the proposed method and those obtained with the numerical analysis are compared. The results show that the proposed simplified method yields reasonably accurate values to those evaluated with a 3D finite difference procedure.

2) Using Eq. (8) the in-plane normal stresses profile in the central concrete slab developed at-the-end of the reservoir filling is calculated. Figure 10 shows the comparison between the results obtained with both procedures. Again, the advanced simplified method estimates quite well the values computed with the 3D finite difference method.

Conclusions

In the design of concrete slabs in CFRDs the stress state induced in the concrete face by the-first-dam reservoir filling loading must be known. To this end, 3D analyses reproducing, as close as possible, the procedures and sequences of construction stages, of dam reservoir filling loading should be carried out. Conscious of the difficulties involved in setting up the numerical models to carry out the analyses required to compute properly the concrete face 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 the Buckingham’s Pi theorem. The proposed method of analysis requires only the sequential use of a series 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. The results obtained with the simplified procedure are similar to those computed with the 3D numerical model. It is important to stress the fact that the height of the CFRD used in the example was different from those employed in the parametric investigation. Accordingly, the results obtained with the advanced simple procedure are reliable and henceforth useful for preliminary designs of the concrete face including the waterstops of actual CFRDs.

According to the analyses presented in this study, normal in-plane stresses increase across the concrete face as the dams become higher, the valleys are narrower, and when the width of the concrete panels increase. These results indicate that special emphasis should be placed on concrete slabs (and waterstops) mainly in the central zone of the concrete face, but also in lateral slabs near the plinth.

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