SEISMIC ANALYSIS OF EARTH WALL GRAVITY DAMS USING DECOUPLED MODAL APPROACH

This study aimed at the assessment of the effects of seismic loads on the earth dam for design purposes. The Unilorin concrete gravity dam is used in comparison with the earth wall gravity dam in seismic zone 1 with a peak ground acceleration of 0.05g. Finite element (FE) method of analysis was used by employing Lagrangian-Eulerian formulation of 4-node plain quadrilateral elements, with modal analysis used to decouple system dynamic equations, thereby saving computational time. The loadings were determined based on EM 1110-2-22000(1995), while the FE model is being implemented using the MATLAB programming tool. Though, the collapse and response of the earth gravity dam is higher than that of concrete gravity dam, the dam satisfies the stability and stress requirements in design.

Keywords: Seismic analysis, earth wall, gravity dam, finite element method 1. INTRODUCTION The structural response of a material to different loads determines how it will be economically utilized in the design process. The loads that structures are subjected to are either static or dynamic, or a combination of both. Static loads are time-independent while dynamic loads vary with time. Dynamic loads can be classified as cyclic, impact or moving but can occur simultaneously. Earthquake is a natural disaster that has claimed so many lives and destroyed lots of property. Earthquake hazards had caused the collapse and damage to continual functioning of essential services such as communication and transportation facilities, buildings, dams, electric installations, ports, pipelines, water and waste water systems, electric and nuclear power plants with severe economic losses. Earthquake is a major source of seismic forces that impinge on structures others are Tsunami, seethe etc. Earth wall is chosen as a material for the dam since its major constituent- earth is abundantly available and provides a sustainable solution.

This necessitates the seismic analysis of concrete gravity dam. Finite element has been widely used in seismic analysis of concrete gravity dams (Waltz 1997, Lotfi 2003) with a defined approach as presented in this programme, using the most natural method based on the Lagrangian–Eulerian formulation.

*appeared in International Egyptian e-journal of Engineering and Mathematics http://www.ieems.net/iejemta.htm, vol.5, 19- Smith (1985) claimed that dams are critical structures that should be made earthquake-resistant, since earthquakes cause severe damages, consequent huge economic and life losses. Hence, the need to prevent the occurrence of these earthquake hazards by carrying out seismic analysis of dams (Polyakov, 1985). Over the years, a lot of work has been done in making earth-fill and concrete dams earthquake-resistant, with advances in structural vibration and finite element method which have aid in the seismic analysis of theses dams. However, not much attention has been paid to the seismic analysis of an earth wall dam. More importantly, most recent methods in seismic analysis of concrete gravity dams have not been employed for the seismic analysis of the earth wall dams. Therefore, the purpose of this is to apply the decoupled modal approach in the seismic analysis of Earth wall dams. Earthquakes had caused severe damages and consequently huge economic losses including losses of lives. On 27th December, 2004, The Punch newspaper reported that Tsunami, a seismic sea wave claimed a lot of lives, rendered many homeless and destitute, and destroyed essential services with Sri Lanka,, Indonesia, Somalia, India, etc been the major casualty countries. Nigeria is not left out from the occurrence of ground motions caused by earth movement as earth tremors had been recorded to occur some recent years back in Abeokuta and Ibadan. Most structures and infrastructure in the country are not earth-resistant.

Also, earth wall, as a sustainable material is quite economical in construction of dams since there is abundant availability of good earth or soil, especially in localities closed to road construction and excavation sites.This study involved the use of finite element method for the dynamic analysis of both the dam and reservoir bodies, using a modal approach as the LagrangianEulerian formulation forming the basis. Also, static loads (weights, hydrostatic pressures) are each visualized as being applied in one separate increment of time. The analytical computation of the modal approach procedure has been carried out and implemented using MATLAB programming tool. The pseudo-static seismic coefficient method was adopted in computing the seismic loads on the dam. The dam used as a case study was assumed to be in seismic zone with seismic coefficient ranging between 0.0 and 0.05. The dam was analysed seismically using the decoupled modal approach and the results were compared with that of the concrete gravity dam.

2. SEISMIC RESPONSE OF EARTH AND CONCRETE GRAVITY

DAMS In earth dams, seismic forces or shaking can induce destabilising deformation or outright failure if not made earthquake resistant. A permanent simplified procedure can be adopted to estimate permanent horizontal displacements of the dams using finite element method that account for non-linear material behaviour and strength reduction due to liquefaction or strain softening. It has been shown ((Hatami, 2001) that the seismic performance of earth dams has been related to the nature and state of compaction of the fill material. Concrete dams structural safety and stability are jeopardized due to the hydrodynamic load of the reservoir that is subjected to ground motion 2.1 FLUID –STRUCTURE SYSTEM During earthquake occurrence, the dam and reservoir body respond differently, as a result of hydrodynamic forces impinging on the fluid body and solid structure. As a result of this, interaction will occur between the fluid–solid structure interfaces as particles move relatively to the mesh points whereas, the meshes moves with the material particles (Bathe, 1996, Qixiang et al. 2000). Much research work has been carried out for the dynamic response of the fluid-solid structure systems. Several methods of analysis for the fluid-structure systems use finite element idealization in the non-linear dynamic response of the system (Fenves and Vargas- Loli, 1988).

ΩF ΩF

Fig. 1.(a) Domains and boundaries of the fluid-structure system In Fig.1(a, b) the following terms are defined as:

= Fluid boundary at reservoir end = Portion of structure boundary along an earthquake = Free surface of the fluid N = Normal to the fluid boundary 3. LOADINGS 3.1 STATIC LOADS.

The static loads are due to (i) The weight of the dam: the unit weight is assumed to be 19.62kN/m3 until an exact unit weight is determined from materials investigation., (ii) Hydrostatic pressure of the water in the reservoir and (iii) The uplift forces caused by hydrostatic pressure on the foundation at the interface of the dam and the foundation. Uplift forces are usually considered in stability and stress analysis to ensure structural adequacy and are assumed to be unchanged by earthquake forces.

3.2 DYNAMIC LOADS Earthquake or seismic loads are the major dynamic loads (Major 1980, Schoeber 1981, Polyakov 1985, Wyatt (1989). being considered in the analysis and design of dams especially in earthquake prone areas. They are of kinematic origin and owe their existence to vibration caused in the structure by the movement of the earth’s surface during an earthquake. They have random characteristics and are regarded as deterministic in practical calculations to simplify the design model. According to EM1110- 2-2200(1995), the earthquake loading used in the design of concrete gravity dams are based on design earthquakes and site specific motions determined from seismological evaluation. The seismic coefficient method is used in determining the resultant location and sliding stability of dams. Seismic analysis of dams is performed for the most unfavourable direction, despite the fact that earthquake acceleration might take place in any direction. Seismic coefficient method of analysis is commonly known, as pseudostatic analysis, and is the ratio of the earthquake acceleration to the acceleration due to gravity. Fig. 2 shows the dynamic loads on a gravity dam. There are different ways of computing earthquake loads on dams. The deterministic approach will be employed where the ground acceleration in terms of g (acceleration due to gravity) is specified for the region where the dam will be constructed.

Hence, the exciting force on the structure is, and where a x, α, g are the ground acceleration, seismic coefficient and acceleration due to gravity respectively.

From Fig. 2 and equation ( ) therefore, the equilibrium system is expressed as:

where Pex, M, ax, W, α, g are the horizontal earthquake force on the dam, mass horizontal earthquake acceleration, weight, acceleration due to gravity and seismic coefficient respectively.

Also Pew, h, te are the additional total water load down to depth y, total height of reservoir, and period of vibration respectively.

3.3 FORCED – DAMPED VIBRATION Forced vibration is the vibration caused by a time- dependent disturbing forced. The governing equation for forced-damped vibration is where M, C, K, Ü, Ů, U, and P(t) are the mass, damping, stiffness, acceleration, velocity, displacement and exciting force on the body.