ارزيابي روش هاي اويلري و لاگرانژي در پاسخ سدهاي بتني وزني با احتساب اندركنش سد-پي-مخزن تحت اثر زلزله

Evaluation of Eulerian and Lagrangian Methods in the Analysis of Concrete ravity Dam Including Dam Water Foundation Interaction under Earthquake

براي فرمول بندي محيط سيال در سيستم سد-پي-مخزن از دو ديدگاه اويلري و لاگرانژي مي توان استفاده كرد كه هر كدام ويژگي هاي خاص خود را دارد. در اين مقاله فرمول بندي اين دو ديدگاه در محيط مخزن سد با در نظر گرفتن اثر اندركنش سد- پي-مخزن به روش اجزاي محدود و اعمال شرايط مرزي مناسب براي سدهاي بتني وزني ارايه شد؛ سپس اين دو روش براي براورد پاسخ سد دراثر زلزله هاي مختلف بررسي شد. نتايج به دست آمده براي دو روش مدل سازي مخزن با تغيير پارامترهاي مختلف [مانند نسبت مدول الاستيسيته جسم سد به پي، شرايط مرزي مخزن، شيب كف و عمق آب مخزن و نيز تاثير رسوبات كف مخزن در جذب امواج الاستيك] از جنبه هاي مختلف مانند دقت و كاهش حجم محاسبات با هم مقايسه شد. نتايج به دست آمده گوياي آن است كه دو روش مدل سازي مخزن از جنبه هاي مختلف مزايا و معايبي دارند؛ اما در مجموع اختلاف قابل توجهي در پاسخ سد به دست نمي آيد.

Hydrodynamic pressure on the upstream face of the concrete dams under the effect of earthquake is one of the most important parameters, in planning damsʹ structure in earthquake zone. Because of the reservoir effect, dynamic analysis of concrete dams is more involved than other common structures. This problem is mostly sourced by the differences between reservoir water, dam body and foundation material behaviors. As a result, researches in this case must be able to evaluate the response of dam with consideration of dam’s interaction with reservoir and its foundation. This problem has been studied vastly by different researchers. The first research on the analysis of concrete gravity dam has been done by Westergaard in 1930 and hydrodynamic pressure on the dam face was obtained by some simplifications. There were a lot of other researches which studied the seismic behavior of the dam-reservoir system, including nonlinear behavior of the dam under pressure and also cavitation. In each research, different modeling methods are presented which are divided into two main groups. In first method which is called Eulerian method, pressure is the main unknown variable in reservoir nodes. In the second method that its main unknown variable is displacement of nodes is called lagrangian method. Each of the methods contain some advantageous and disadvantageous. The purpose of this paper is to evaluate possible advantages and disadvantages of both methods. Specifically, application of the above methods in the analysis of dam-foundationreservoir systems is leveraged to calculate the hydrodynamic pressure on dam faces. Within the frame work of dam- foundation-reservoir systems, dam displacement under earthquake for various dimensions and characteristics are also studied. To achieve this purpose, visual C#.NET 2003 computer programming language is used in this investigation that produces possibility of dynamic analysis of concrete dams under earthquake with system modeling by both methods. Nine node elements for reservoir and eight node elements for dam and foundation are used for both methods. Also newmark average acceleration method is used for solving dynamic’s equilibrium equation. Modares Civil Engineering Journal (M.C.E.L) Vol.11, No.4, Winter 2011 131 In this paper the response of the tallest, non-overflow monolith of Pine Flat dam in California, which is 122 m high, to horizontal and vertical component of earthquake is computed. A water depth of 116 m is considered in full reservoir condition, and the water has the following properties: unit mass,? ? 1000 kg /m3 , bulk modulus, K ? 2.07*109 kg/m2 , and pressure wave velocity, w c ?1440m/ s . The finite element model of reservoir consists of 12 isoparametric elements and it extends upstream a distance of 366 m, three times the dam height. The dam consists of 20 isoparametric elements. The concrete of dam has the unit mass of? ? 2500kg /m3 , young’s modulus ofE ? 2.275*1010kg /m3, and poisson`s ratio of ? ? 0.25. The concrete of foundation has the unit mass of , young’s modulus ofE 4.45*1010 kg / m3 f ? , and poisson`s ratio of ? 0.25 f ? . The peak acceleration of S69E and vertical components are 0.18g and 0.1g, respectively. The results of both Lagrangian and Eulerian methods for Pine Flat dam are quantitatively evaluated and compared in different condition and following results are achieved: 1- In Lagrangian Method, there is only one variable in equilibrium equation and mass and stiffness matrixes are symmetric. But there is not such a condition in Eulerian method. Also, the numbers of unknown parameters are different in two methods. By considering these differentiations, needed time for analysis of Pine Flat Dam under Taft earthquake ,with mentioned characteristics, by Lagrangian method is 1.17 times more than needed time for Eulerian method. 2- The effect of material on reservoirʹs bottom in absorbing energy and `reducing systemʹs response was considerable especially under vertical component of the earthquake. Results indicate that this case is not affected by reservoir modeling method. By applying this effect, the response will decrease about 15% under horizontal component and 60% under vertical component of the earthquake. 3- By evaluating the effect of reservoir bottomʹs slope, it is concluded that in the case of rigid foundation, the response by Lagarngian modeling is about 10% more than Eulerian one. With the increase of slope, the response will decrease under horizontal component of earthquake but it will decrease or increase about 13% under vertical component of the earthquake. In other words, reservoir bottom slope has little effect on response of the system under both vertical and horizontal component of earthquake. But this effect is not negligible. 4- It is included from the analyses that by decreasing the depth of reservoir the response will decease up to 50 percent under horizontal component of earthquake. This amount is 80 percent under vertical component. Also in the case of decreased depth, response of Lagarngian method is about 10% more than Eulerian method. 5-In all analysis, the assumption of rigid foundation results in greater answers than the cases of flexible foundation.