Development of a practical methodology for the analysis of gravity dams using the non-linear finite element method

Abstract

In the classical design method for gravity dams, the designs are conducted in the linear elastic isotropic material domain. For many decades the so-called ‘classical method’ (or conventional method) was used to design gravity dams. This method is based on the Bernoulli shallow beam theory. Despite much criticism expressed by academics regarding the basis of the theory, dam design engineers are still using the classical method to design gravity dams. Currently, in most dam building countries the various codes of practice are standardised and based on this method, and engineers have confidence in these codes. This state of affairs will probably continue until structural engineers come up with a viable alternative for designing gravity dams more precisely. The perception of increased risk is always a critical aspect to overcome when introducing an alternative design method, especially when the established, well-known methodology has proved to be safe. However, when so-called ‘back analyses’ are performed on existing dams, it is not so straightforward to assess the safety margin of these structures. Material properties and their yielding or failure characteristics are now becoming important in evaluating these structures accurately in the non-linear domain. With the growing popularity of roller compacted concrete as a dam building material, the attractiveness of gravity dams has also increased and the author is of the opinion that the finite element method could be utilised more efficiently to optimise gravity dams. But, as with any new or alternative design method, the parameters and means of evaluation should be developed to establish a workable and reliable technique. The objective of this dissertation is to develop a practical methodology for the non-linear analysis of gravity dams by means of the non-linear finite element method. A further aspect of this dissertation is the inclusion of a broad guideline on the application of the latest dam design standards used in South Africa for both the classical and finite element methods. In order to gain a better understanding of the basic design criteria, a literature survey was conducted on the evolution of dams and the various theories developed in the past to design and optimise gravity dams. The literature survey included the examination of gravity dam safety criteria and some available statistics on dam failures. The International Committee on Large Dams (ICOLD) has interesting statistics on dam failures and their causes. A few typical dam failures are presented to illustrate what can go wrong. During this literature research, a thorough study was done on the non-linear material theory, with special reference to the Mohr-Coulomb and Drucker Prager material models. The findings of the study are used to illustrate how the non-linear material models are incorporated into the finite element method and in what manner the different material parameters have an influence on the accuracy of the results. As already mentioned, currently the classical method is still a recognised design standard and for this reason a summary is presented of the South African Department of Water Affairs and Forestry’s practice for designing gravity dams. This includes the latest concepts on load combinations and factors of safety for these load conditions. This summary of current practice is used as a stepping stone for the proposed load combinations that could be used for the finite element method as these are not always compatible. However, this dissertation does not deal with the full spectrum of load combinations and the scope is limited to hydrostatic loads. Although the finite element method is a very powerful structural engineering tool, it has some serious potential deficiencies when used for dam design. The most serious problem concerns the singularities and mesh density, which develop high stress peaks at the heel of the dam wall. This problem is illustrated and some practical finite element examples are given. Some solutions for addressing this problem are also presented. It is concluded that an effective method for overcoming the singularity problem is to use the non-linear material yielding model. In order to calibrate the non-linear Drucker Prager model, several finite element benchmarks were conducted, based on work done by other researchers in the fracture mechanics field. Although the theory of the Drucker Prager model is not based on fracture mechanics principles, this model simulates the failure of the concrete material very well. To demonstrate this, various benchmarks were conducted, such as a pure tension specimen, a beam in pure bending, a beam combined with bending and shear, the flow models of Chen (1982), a model of a gravity dam and, finally, a full-size gravity dam. The next step in the study was to calibrate the Drucker Prager model with the concrete material properties used in existing dams of different construction methods and ages. The material strength of the concrete was statistically analysed and the average strength was calculated. The important ratio of tensile strength to compression strength (ƒt/ƒc) was also examined and the findings are presented. This ratio is important to get accurate results from the Drucker Prager model. The normal input parameters for the Drucker Prager model are the internal friction angle of the material (φ) and the cohesion (c). Scrutiny of the work done by Chen (1982) helped to find a useful solution to obtaining the parameters for the non-linear finite element method without determining the ö and c values, but by using the material tensile and compression strengths instead. The formulation is demonstrated in the chapter on theory. To illustrate the usefulness of the non-linear yielding model a few case studies were conducted. A hypothetical triangular gravity dam structure was analysed because it was widely used in other literature studies and a useful comparison could be made. Then, a case study of an 80-year-old concrete gravity dam was performed. The uniqueness of this dam lies in the fact that it was designed before the theory of underdrainage was used in South Africa and the dam has a characteristic shape due to its relatively steep downstream slopes compared with today’s standards. A study of material strength sensitivity was also done on this dam to evaluate its stability under severe load conditions. The last case study presented is on a recently designed 75-m-high roller compacted concrete gravity dam, optimised primarily by the classical method. The non-linear Drucker Prager yield model was used to evaluate this structure, with the actual material strengths taken from the laboratory design mix results. Although the finite element method was used during the design stage of this dam, it was used mainly to check the results of the classical method. The finite element method was also used to do studies on this dam where the classical method could not be used, such as studies of temperature and earthquake load conditions (not included in this research). The factor of safety against sliding was also determined using the results obtained from the finite element method and compared with the results obtained from the classical method. This case study gives an approximate comparison between the classical method and the finite element method. Finally, a methodology is proposed for analysing a gravity dam. Procedural steps are given to describe the methodology. With regard to the future, the advantage of the non-linear finite element method is that it can easily be extended to contemporary 3-D analysis, still using the same concept. Many dams can only be accurately evaluated by a full 3-D analysis. There is a modern tendency to design gravity dams in 3-D as well so as to evaluate their stability against sliding in the longitudinal direction. The non-linear 3-D finite element method is also used for arch dams, for which very few alternative numerical analysis methods are available. Moreover, the non-linear finite element method can be extended to earth and rock-fill embankments.