As shown in an example analysis of Pine Flat Dam (linear dam), cavitation occurs in the upper part of the reservoir along the dam. Weak planes in the foundation can affect the dam in different ways [3,4] depending on their profile and load-bearing capacity.
Scope and Organization
The eminent threat posed by weak aircraft in the foundation may have been demonstrated by the failure of the Malpasset Dam in France [3]. A two-dimensional treatment is also often used where unkeyed contraction joints are present in the dam, on the grounds that the monoliths will vibrate almost independently under vigorous shaking; A flat stress state is usually assumed.
Scope and Organization
Mathematical Formulation
Concrete Gravity Dam
In the context of crack opening in concrete, a special treatment of Rayleigh damping is required, as will be discussed later in Chapter 4. Following standard procedures, the displacements in the interior of an element can be obtained using the element interpolation functions and the element strains evaluated from the appropriate derivatives of the element displacements.
Rock Foundation
For the purpose of calculating the foundation interaction coefficients, the foundation surface in contact with the dam is assumed to be a rigid plate of the same dimensions as the dam base (Leq x W, where W is the base width of the dam). Foundation interaction is incorporated by pooling Ket and C01 into the dam stiffness and damping matrices, respectively, at the selected three nodal degrees of freedom at the dam base.
Water
However, in the displacement-based formulation, it should be included to help suppress the property of the water stiffness matrix. Comparison of volume strain time histories between the current formulation (solid line) and the exact solution (dashed line) for the test problem.
Earthquake Response
A stiffness proportional damping was added to the water to give 0.1% damping in the water fundamental pressure state. In the finite element discretization, a length of the reservoir corresponding to the water depth is meshed. A nonlinear finite element analysis of the 246 ft tall Norris Dam [1] included nonlinearity in compressive stress, hysteresis, and smeared cracking (critical tensile stress).
Time histories of the vertical component of the normal stress at the crack level (elements 73 and 80) for problem 4.1. Time histories of absolute volume strain in water along a cavitation-allowed dam wall. The upward fall of the crack may be due to the initiation at the lower part.
Time history of the displacements at nodes 1, 130 and 139 in the direction tangent to the crack for Problem 4.2. The results of problems 4.1 and 4.2 showed that the upper part of the dam tends to shift in the downstream direction. Time history of the displacements at nodes 139 and 148 in the direction perpendicular to the crack for problem 4.3.
Time histories for the displacements at nodes 1, 139 and 148 in the direction tangential to the crack for task 4.3. Time histories for the displacements at nodes 156 and 177 in the direction perpendicular to the crack for task 4.6. Cracking of the dam reduces the dynamic water pressure generated in the reservoir and thus the amount of cavitation.
Introduction
A major problem in modeling tank cavitation is the lack of knowledge of the actual physical mechanism involved. The behavior of the model is investigated in Section 3.5 with emphasis on element size effects.
Previous Work
In case of strong excitations, these separations extended over the upper third of the dam face and were followed by compressive pressures upon impact. The effects of cavitation on dam response were moderate and generally slightly reduced peak responses.
Mathematical Formulation
3.1) where pe is the absolute water pressure at a point in element e; gij is the corresponding volume tension given by. The reshaping of the vector of the stiffness forces, pk in equation (2.1), is done at the element level.
Numerical Studies
It can be seen that the cavitated elements become larger and undergo high-frequency oscillations after the cavitated region is closed. The damping constant, bw, should be chosen so that most of the spurious components dissipate without significantly affecting the "true" response. The results of finer meshes with the same amount of damping (ie, the same weight) are shown in Fig.
Note that spurious components are more effectively eliminated by finer meshes due to the use of stiffness-proportional damping. While this was not attempted due to expense, it is believed that such a result would differ little from the responses in fig. Even with small damping, such forces can become significant if the volume strain rate is high.
On the other hand, too little damping can result in very unrealistic cavitation volumes due to spurious oscillations. It is worth noting that energy radiation through the transmission boundary at the upper end of the reservoir does not provide adequate means to dissipate spurious components.
Cavitation Effects on Dam Response
3.11, the degree of cavitation is limited to the first and second column of elements upstream of the dam. Its removal was achieved by increasing the stiffness-propodional damping in the water to 3% of the critical value at the fundamental pressure; results appear in figures. The figures show that the 3% damping in the water is still low enough not to significantly affect the fundamental components of the system response.
The dominance of the hydrodynamic response of the high-frequency components observed in [7] and [3] when cavitation was included is apparently due to their use without internal water damping, or a much smaller amount than used here. Also, their more extensive upstream propagation of cavitation is probably possible attribute the same characteristics. This is in qualitative agreement with the results obtained here with stiffness-proportional damping in water on the order of 1% to 3% of the critical in the fundamental pressure mode.
F., "Study of the earthquake response of Pine Flat Dam", Earthquake Engineering and Structural Dynamics, Vol. In the first case, finite element cracks are smeared out by adjusting the element constitutive description for open and closed crack cases.
Scope and Organization
Previous Work
- Field and Laboratory Experience
- Mathematical Studies
Concrete flow at the contraction joints indicated that there was some individual vibration of the monoliths. A repetition of the 1.21 g test on the cracked dam caused some damage to the downstream end of the crack due to impact, but stability was maintained. A horizontal cut all the way through the dam was made at the height of the observed fracture to represent a pre-existing fracture.
A single monolithic model of the Koyna Dam showing cracks from a shaking table test at the University of California, Berkeley [16]. Due to the many assumptions, some of which are questionable, used in the analysis, the results should be treated with great caution. During the earthquake analysis, almost all surface elements cracked, as well as most of the internal elements in the upper third of the dam.
All the crack zones spanned at least several integration points in the vertical direction and often several elements. Third, the satisfaction of the crack criterion in a number of elements is simultaneous, that is, in an iteration of a time step (here avoided by imposing a limit of one new crack per iteration).
Crack Modeling
- Open Crack Representation
- Closed Crack Representation
- Crack Initiation
- Crack Propagation
- Planes of Weakness
However, this results in an undesirable zero normal stiffness for fibers oriented parallel to the crack. This is shown in Appendix 4.B for a rectangular 4-node element when the crack is not oriented parallel to one of the element sides. In the latter procedure, the crack is introduced into the element for which the ratio Upt to Uct is maximum.
To adequately represent such behavior in the finite element idealization, the mesh must be fine with respect to the size of the fracture zone. When the principal tensile stress in the element just before the crack front exceeds the critical element strength given by. Candidates for crack tip extension in an early, automated version of the propagation algorithm.
The current version of the computer program allows the user to control the crack extensions through interactive programming. The current version of the computer program takes into account areas of weakness in the dam, such as construction joints and initial cracks.
Numerical Analyses
- Problem 4.1
- Problem 4.2
- Problem 4.3
- Problem 4.4
- Problem 4.5
- Problem 4.6
First, consider the simple problem of an incipient horizontal crack near the top of a dam. Despite the release of tensile stresses normal to the crack (Fig. 4.12), the contours of the maximum principal stress (tension) shown in Fig. The only other stress large enough to initiate cracking occurs at the fifth dam, as shown in Fig.
Dam rupture also affects the hydrodynamic pressures in the reservoir and consequently the amount of cavitation. Time histories of the vertical component of the normal stress above and below the crack for problem 4.1. Tensile stresses high enough to re-initiate additional cracks occur near the crack on a plane almost perpendicular to it and also at the fifth dam (Figure 4.28).
These stresses close to the crack occur at the downstream end due to the upstream dip of the crack. The contours of the maximum principal stress (Fig. 4.33) show patterns similar to those observed in the previous exercises. Vertical stress developed in the upstream elements at the base of the neck, but never peaked at 600 psi during this cycle.
Deflections of the dam toward the reservoir are particularly significant, and in the largest excursion the inclined crack remains open at 0.
![Figure 4.3. A single monolith model of the Koyna Dam showing cracks from a shaking table test at the University of California, Berkeley [16]](https://thumb-ap.123doks.com/thumbv2/123dok/11054062.0/100.922.243.781.109.687/figure-single-monolith-showing-shaking-university-california-berkeley.webp)
