Hazard potential of zones of weakness in gravity dams under impact loading conditions

Abstract

Dam constructions worldwide are designed and constructed in view of the strictest safety aspects for all static and dynamic load cases. As experience shows, however, formation of cracks in the “homogeneos concrete” as well as unsatisfactory compound behavior of lift joints are not to be excluded. These zones of weakness especially on the upstream side of the dam— exposed to high water pressure (static and dynamic)— represent an increased risk of safety. The main investigation, apart from the computation of the dynamic effects on the dam as a global structure, focuses on the stability analysis of a pressure-water filled crack configuration subjected to “dynamic loading” in the form of seismic action on the dam-reservoir-system and alternatively by “impact spot-loading” within sectors of the reservoir. A fracture mechanics based analysis shows an excessive potential of damage for the afflicted structure.

Keywords

dam-reservoir / earthquake / impact / cracking / fracture mechanics

Introduction

In analyzing the global stability (sliding, overturning), a concrete gravity dam usually is considered as a rigid continuum with stability assessments via the equilibrium of overall forces acting on the body. This approach is the main emphasis in the design criteria for dams with high standards of safety. However, sophisticated analysis techniques are increasingly used for the investigation of local effects, e.g., cracking, taking into account all the aspects associated with it. Cracks like defects in concrete dams always have been a matter of concern, as shown in the extensive literature on that theme. In comparison with cracking in building constructions (structural engineering), a much higher potential of risk is given by water pressure action on crack endangered structures. Whereas the water penetration into joints or cracks in the pure static case (including the “pseudo-dynamic” Westergaard pressure-profile) is characterized by the hydrostatic pressure and therefore the investigation is restricted to the pure dam structure, in dynamically excited dam-reservoir systems the fluid has to be modeled. Dynamic impacts in a reservoir may be triggered by earthquakes, rock-falls, abrupt landslides (reservoir banks), sudden geological fault movement, and last but not least, terrorist attacks.

The main investigation, apart from the computation of the dynamic effects on the dam as a global structure, focuses on the stability analysis of a pressure-water filled crack configuration subjected to “dynamic impingement” in the form of a seismic action on the dam-reservoir-system (Part 1) and alternatively by “impact spot-loading” within certain sectors of the reservoir (Part 2).

Earthquake investigations concerning dam-reservoir-systems are well established, and to a high degree of application, developed and comprehensively embodied in standards.

The effect of impact-generated pressure waves on the dynamic interaction between the reservoir and the dam may be considered significantly different when compared to the strong-motion earthquakes. Initiation areas on the surface, within and on the bottom of the reservoir could be assigned to special triggering mechanisms:

● Near-surface bursts may be classified as kinetic energy penetrators which can be caused by rock-falls, abrupt landslides (reservoir banks), etc.

● Underwater blasts (UNDEX) are categorized as chemical energy penetrates. The explosive-impingement contact area is characterized by the formation of a significant shock-wave with all its side effects as bubble formation, cavitation, etc. This case is not taken into consideration here.

● Reservoir bottom trigger mechanisms may occur in a kind of uncontrolled dynamic interaction between the reservoir and a structural discontinuity such as a fault in the foundation. When the frictional resistance of a fault is suddenly removed the elastic energy stored in the vicinity of the fault is released by an unstable high-speed stress wave’s propagation with interaction with the reservoir foundation interface.

Fracture mechanics methods offer the possibility to understand the stress-field in the crack tip region. The time dependant stress-intensity-factor-behavior at the crack tip serves as an assessment of the hazard potential for the overall structure.

Object of investigation

The object of the investigation is related (refers) to a dam-reservoir-system with a gravity dam—afflicted with a horizontally spreading crack on the upstream side—and a reservoir with a rectangular cross section and 600 m extension in length (RL) in the case of the earthquake investigation (Part 1), and 200 m of length in the impact analysis (Part 2), respectively (see Fig. 1).

The main study focuses on the determination of the SIF-Limit (fracture criterion) representative for the stability behavior of the crack under dynamic impingement of the system.

Methodology

The present investigation essentially refers to two special engineering fields. As a first aspect, both the dynamic behavior of the dam-reservoir-system (Part 1) and the transmission of the impact generated pressure waves (Part 2) have to be modeled (analyzed) and, in further consequence, the resulting effects on the stability behavior of a fluid filled crack in the dam have to be considered.

The analysis was carried out with SOLVIA-Finite Element-System SOL03 developed by SOLVIA Engineering AB, Västeras, Sweden. Besides all features representative for present-day finite element routines in engineering, version SOL03 includes 2D and 3D fluid elements with displacement and potential based formulations respectively. The structure-fluid system is represented by 8-node plane elements and 9(8)-node potential based fluid elements. The fluid-structure interface is established automatically without any user specification. The crack tip area is modeled with a mesh-rosette of quarter-point elements for a strain-energy-release and a displacement based evaluation of the modus-I and modus-II stress-intensity-factors (SIF-K_I and K_II).

Fluid representation

The fluid element formulation is deduced from the Navier-Stokes equation (1) under presumption of small motions (neglecting the convective terms) and an “inviscid” fluid characteristic (disregarding viscous effects). Under consideration of continuity and constitutive relationships (2) the typical dynamic specification (3) for an “acoustic medium” is obtained:

(1)\rho {\displaystyle \frac{\mathrm{\delta }\mathit{u}}{\mathrm{\delta }t}}=-\nabla p,

(2){\displaystyle \frac{\mathrm{1}}{K}}{\displaystyle \frac{\mathrm{\delta }p}{\mathrm{\delta }t}}=-\nabla ·\mathit{u},

(3){\nabla }^{\mathrm{2}}p={\displaystyle \frac{\mathrm{1}}{c^{\mathrm{2}}}}\ddot{p},

where BoldItalic is the fluid velocity, ρ is the density, p is the dynamic pressure, K is the bulk modulus of the fluid and c=\sqrt{K/\rho }, the wave celerity.

In SOLVIA, a velocity potential Ф (with adequate boundary conditions on S) is introduced in Eq. (4):

(4)\begin{array}{lllll}\mathit{u}=\nabla \mathit{\Phi },\hfill & p=-\rho \dot{\mathit{\Phi }},\hfill & {\nabla }^{\mathrm{2}}\mathrm{\Phi }={\displaystyle \frac{\mathrm{1}}{c^{\mathrm{2}}}}\ddot{\mathit{\Phi }},\hfill & {\displaystyle \frac{\mathrm{\delta }\mathit{\Phi }}{\mathrm{\delta }n}}=u_n^s,\hfill & -\rho \dot{\mathit{\Phi }}=p^s.\hfill \end{array}

To consider various processes, characteristic for this coupled system—initial and boundary terms as, e.g., interface and base displacements, free-surface displacements (development of surface waves acting as an equivalent spring), external surface pressure, initial pressure and especially the gravity pressure (p_g = ρgz_g) — energy principles have been used.

Fracture mechanics aspects—cementitious materials

The technological development in the field of fracture mechanics of cementitious materials—in terms of material characterization and analytical/numerical simulation—offers the possibility of undertaking a realistic investigation in this field of engineering, as has been started off by Hillerborg et al. [1]

Generally, fracture mechanics methods via appropriate fracture criteria (linear and/or nonlinear concepts) may offer possibilities for the investigation of such (cracked) structures. The term “appropriate” primarily is to understand the concept in connection with the availability of a “valid” fracture mechanics material parameter for the problem to be investigated. The requirement, however, is particularly difficult to meet in case of mass concrete where the size of the aggregates can often spread up to 120 mm and across.

Apart from the already often tested analytic-numeric methods both in the linear elastic range of application (linear elastic fracture mechanics (LEFM)) and in the nonlinear mode of application (smeared crack-damage mechanics concepts), therefore, the determination of adequate fracture mechanics material parameters (fracture toughness, fracture energy) for mass concrete is of great importance. New testing methods based on wedge-splitting samples for normal concrete and mass concrete have been developed by Linsbauer et al. [2]

Within this contribution linear elastic fracture mechanics concepts are applied—a legitimate assumption due to the small ratio of the fracture process zone to the global dimension of the dam structure (see Fig. 2).

The LEFM-representative fracture mechanics parameter stress intensity factors (K_I, K_II, K_III) and energy release rate are determined via the triangular-quarter-point element displacement relationship as originally proposed by Barsoum [3].

The aforementioned methods also successfully have been applied for cracking investigations in dams as, for example, may be seen in Figs. 3 and 4.

The investigation was carried out by the author with FEFAB (pre-version of FRANC2D), a two dimensional fracture mechanics routine developed at the Cornell-University in Ithaca (Fracture Group Ingraffea). The FEM crack propagation study compares quite well with the actual cracking path and also serves as contribution for repair measures.

Rounding off this section, two typical fracture mechanics criteria characteristic for the LEFM are cited:

The fracture mechanics analysis is directed to the determination of critical fracture mechanics parameters for the assessment of the “stability behavior” of the (cracked) dam. In particular, the critical “phase” of the dynamic loading spectrum for the maximum crack-opening (maximum crack-fluid-pressure) must be filtered out and the associated values of the stress intensity factors K_I and K_II are “confronted” with K_IC (fracture toughness)—value representative for the material.

Wedge splitting tests carried out by Linsbauer [4] on drilling cores of the Austrian Koelnbrein Dam showed a mean K_IC of 2.38 MPa·m^1/2.

Analysis—Part 1 (strong motion earthquake)

The earthquake investigation of the dam-reservoir system in a first step requires the estimation of the frequency-range of the system via a complex-harmonic-analysis to point out the critical phase of the exciting impact on the structure.

The finite-element mesh configurations of the dam-reservoir configuration and the crack-tip-block are seen in Figs. 5(a) and (b). For simplification—but without disruption of the general concept of this investigation—a rigid foundation and a non-radiation boundary condition at the end of the (600 m length) reservoir are postulated.

The acceleration response at the upstream-side crown edge as a function of the excitation frequency is seen in Fig. 6. The numerical values of the characteristic frequencies are pointed out in Table 1.

The numerical values of the characteristic frequencies are pointed out in Table 1 both for the empty (Dam & Air- Modes 1 to 5) and full (Dam & Res- Modes 1 to 10) reservoir. The lowest natural frequency of the dam in air is 4.53 Hz and is lowered to 3.46 Hz in the case of fluid-structure interaction (full reservoir).

For the dynamic study of the dam-reservoir system, the El Centro-NS earthquake spectrum shown in Fig. 7 was used and it indicates a PGA (peak ground acceleration) value of 0.35g at a time of 2.12s. For the present investigation the duration of the earthquake was reduced to 15 s.

The El Centro earthquake response spectra at the upstream side crown-point are pointed out in Fig. 8.

The peak-frequencies show up correspondence with the natural Mode 1 and Modes 2, 5, 9 and 10 of the dam-reservoir system marked in Table 1 (in bold outline). The pressure distribution at the time of 9.567 s is seen in Fig. 9.

Results of fracture mechanics analysis (Part 1)

The effect of the El Centro spectrum with a PGA value of 0.35g yielded an excessively high value (at time of 9.57 s duration) for the stress intensity factor, indicating a progressive crack propagation with serious consequences for the overall stability of the dam.

To locate the threshold value for the stability of the crack, two additional investigations with PGA values of 0.2g and 0.1g for the earthquake spectrum of El Centro were carried out. The results are listed in Table 2.

Neglecting the insignificant values of K_II and keeping in mind the threshold material value of K_IC = 2.38 MPa·m^1/2 the only value “within touch” is the one associated with a PGA value of 0.10g (aside from meeting any safety demands).

Analysis—Part 2 (impact loading within the reservoir)

The investigation of this type of reservoir impingement calls for significant alternative strategies in comparison to the earthquake investigation. Particularly, the configuration of the FE-mesh requires special consideration concerning the adequate modeling of the pressure (shock) wave propagation.

In principle, the element size in reproducing impact-wave-propagation is closely connected to the pulse-duration and should cover the complete pressure-wave characteristic. In most of such cases “explicit time integration methods” are used with quadratic 4-node fluid elements (lumped mass matrix). In SOLVIA the potential based Fluid2D and Fluid3D-elements require implicit methods (e.g. Newmark, Wilson, Hilber). In the present case a 200 m section of the reservoir with 100 m water depth has been generated with 8-node Fluid2D elements of the size of 1.0 m to 1.0 m, which is seen in Figs. 10(a) and (b).

The loading configuration and the pressure wave development (20 ms) due to an impact pulse with 15 ms duration and a maximum value of 20.0 MPa acting on the surface (e.g. rock-fall) and on the bottom (e.g. fault asperity “brittle ruptur”) respectively, are shown in Figs. 11(a) and (b).

Results of fracture mechanics analysis (Part 2)

Time periods between 1.0 ms to 100 ms are filtered out and the associated values of the stress intensity factors K_I and K_II are calculated. These characteristic parameters are “confronted” with the K_IC (fracture toughness) value, representative for the material, via the failure assessment diagram (FAD) for mixed mode conditions used herein, which is represented in Fig. 12(a). The associated impact pulse characteristic is shown in Fig. 12(b).

The results may generally be assigned to three different states: crack closed with K_I = 0, crack open and stable (S 10 MPa) and crack open and unstable, as shown by the failure assessment diagram (see Fig. 12(a)).

For all load intensities p₀ up to a maximum pulse of 5 MPa, the crack is closed. This is basically due to the fact that the design was orientated in view of a more compact geometry with a ratio of m = BW/H = 0.81, which in structural (beam) design analysis results in compressive stresses on the upstream side of the dam to a depth of approximately 2/3 of the dam height, even under consideration of water penetration into a lift-joint (levy profile).

From 5.0 MPa intensity up, only a small margin is left for a safe phase, because at more than 10 MPa, especially in the case of the bottom impact, the situation already is on the borderline of stability (quite apart from meeting any safety demands), as may be seen in Fig. 12(a).

Conclusion

The present investigation was directed to the identification of characteristic fracture mechanics parameters representative of the stability behavior of a crack on the upstream side of a dam subjected to earthquake and “shock” loading. In particular, a gravity dam with 105 m height and a postulated crack of 5.0 m in length spreading horizontally from the upstream-side of the dam at a depth of 60 m was taken as subject for the analysis.

The outcome drastically demonstrates the hazard potential of such a configuration, already starting from an El Centro type earthquake reduced to one third of the max PGA and a pulse intensity of 10 MPa in the case of an impact event within the reservoir.

The question arises whether a postulated crack exposed to full water pressure under dynamic loading must be taken into account even in the initial stage of the design process.

Remark This contribution partly may be seen as summary form of the investigations already presented in the LTESBD08 [5] and LTBD09 [6] conferences and proceedings, broadened with additional comments. Extensive references can be found there.

References

[1] Hillerborg A, Modeer M, Petersson P E. Analysis of a crack formation and crack growth in concrete by means of Fracture Mechanics and Finite Elements. Cement and Concrete Research, 1976, 6(6): 773–782

[2] Linsbauer H N, Tschegg E K. Fracture energy determination of concrete with cube shaped specimens. Zement und Beton,1986, 31: 38–40

[3] Barsoum R S. Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements. International Journal for Numerical Method in Engineering, 1977, 11(11): 85–98

[4] Linsbauer H N. Fracture mechanics material parameters of mass concrete based on drilling core tests –review and discussion. In: van Mier J G M, Rots J G, Bakker A, eds. Fracture Processes in Concrete, Rock and Ceramics, RILEM. London: E. & F.N.Spon,1991, 779–787

[5] Linsbauer H N. Dynamic pressure wave action in zones of weakness in dams – triggered by earthquake-impact-loading in the reservoir. In: Zhu Y M, Liu S H, Qiang S, Chui A, eds. Proceedings of the 1st International Conference on Long Time Effects and Seepage Behavior of Dams (LTESBD08), Nanjing. Nanjing: Hohai University Press, 2008, 112–121

[6] Linsbauer H N. Damage potential of an upstream-side crack in a gravity dam subjected to an impact loading in the reservoir. In: Bauer E, Semprich S, Zenz G, eds. Proceedings of the 2nd International Conference on Long Term Behavior of Dams (LTBD09), Graz. 2009, 841–846

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