지반보의 응력-변형률 거동에 대한 해석법 비교

이승현(Seung-Hyun Lee),한진태(Jin-Tae Han) 한국산학기술학회 2018 한국산학기술학회논문지 Vol.19 No.12

간극수압을 받는 지반보의 응력-변형률 거동 분석을 위해 해석해와 유한요소해석결과를 정량적으로 상호 비교해 보았다. 유한요소해석을 통해 얻은 수평응력은 해석해에 의한 결과와는 달리 지반보의 수평축에 대하여 대칭성을 보이지 않았으나 요소의 개수가 증가함에 따라 대칭에 가까운 형태를 보였다. 해석해에 의한 수평응력을 유한요소의 가우스점에 대하여 얻은 수평응력과 비교해 볼 때 3개의 요소를 고려한 유한요소해석을 통해 얻은 인장응력의 값은 해석해에 의한 최대 인장응력값의 6% 였고 압축응력의 값은 해석해에 의한 최대값의 37% 였다. 6개의 요소를 고려한 유한요소해석을 통해 얻은 인장응력의 값은 해석해를 통해 얻은 최대값의 61% 였고 압축응력의 값은 해석해를 통해 얻은 최대값의 83% 였다. 지반보내에 발생되는 연직응력은 해석해에 의할 경우 보의 깊이에 따라 연속적인 분포양상을 보인다. 유한요소해석에 의한 연직응력은 유한요소를 구성하는 요소에 따라 이산적인 분포를 보이는데 요소내의 4개의 가우스점에 대하여 얻은 평균 연직응력은 지반보에 작용하는 간극수압의 크기에 가까운 값을 보였다. 지반보의 중앙 근처에서의 연직변위량을 비교해 볼 때 3개의 요소로 구성된 지반보에 대한 유한요소해석을 통해 얻은 값은 해석해에 의한 값의 35% 였으며 6개의 요소로 구성된 지반보에 대한 유한요소해석을 통해 얻은 값은 해석해에 의한 값의 57% 였다. To analyze the behavior of a soil beam under pore water pressure, the results of analytical solutions and finite element analysis (FEM) were compared quantitatively. In contrast to the results of the analytical solution, the horizontal stress obtained from the FEM did not show a symmetrical distribution. On the other hand, the horizontal stress became closer to symmetrical distribution as the number of elements of the soil beam were increased. A comparison of the horizontal stresses from the analytic solution with those obtained from Gaussian points of FEM showed that the magnitude of the tensile stress from the FEM using 3 elements was 6% of the maximum value of the analytical solution and the compressive stress from the FEM using the same elements was 37% of the maximum value of the analytical solution. The magnitude of the tensile stress from the FEM using 6 elements was 61% of the maximum value of the analytical solution and the magnitude of the compressive stress from the FEM using the elements was 83% of the maximum value of the analytical solution. Vertical stresses, which were obtained from the analytical solution, showed a continuous distribution with the depth of the soil beam, whereas the vertical stresses from the FEM showed a discrete distribution corresponding to each element. The results also showed that the average value of the vertical stresses of each element was close to that of the pore water pressure. A comparison of the vertical displacements computed at the near vertical center line of the soil beam from the FEM with those of the analytical solution showed that the magnitude of the vertical displacement from FEM using 3 elements was 35% of the value of the analytical solution and the magnitude of the vertical displacement from FEM using 6 elements was 57% of the value of the analytical solution.

Size distribution of polydispersed aerosols during condensation in the continuum regime: Analytic approach using the lognormal moment method

Jung, Chang H.,Park, S.H.,Kim, Yong P. Elsevier 2006 Journal of aerosol science Vol.37 No.10

<P><B>Abstract</B></P><P>An analytic solution for polydispersed aerosol condensation was obtained in the continuum regime. In this new approach, lower order moments, as compared to a previous study, were used to the log-normal aerosol size distribution, in order to obtain an analytical solution. The resultant analytic solution based on the lognormal size distribution was compared with the exact solution and proved to be in good agreement. The obtained solution was also found to be consistent to a greater degree with the exact solution was the previous analytic moment solution for the polydispersed aerosol condensation process. This is particularly true for cases in which broad size distributions exist.</P>

시멘트 페이스트의 슬럼프 유동 모사를 위한 분석적 해의 검토

윤태영 한국도로학회 2016 한국도로학회논문집 Vol.18 No.3

PURPOSES : In this paper, the analytical solutions suggested to simulate the behavior of rheological fluids were rigorously re-derived and investigated for fixed conditions to evaluate the applicability for the solutions on a mini-cone slump test of cement paste. The selected solutions with proper boundary conditions can be used as reference solutions to evaluate the performance of numerical simulation approaches, such as the discrete element method. METHODS: The slump, height, and spread radius for the given boundary and yield stress conditions that are determined by five different analytical solutions are compared. RESULTS: The analytical solution based on fluid mechanics for pure shear flow shows similar results to that for intermediate flow at low yield stresses. The fluid mechanics-based analytical solution resulted in a very similar trend to the geometry-based analytical solution. However, it showed a higher slump at high yield stress and lower slump at low yield stress ranges than the geometry-based analytical model. The analytical solution based on the mini-cone geometry was not significantly affected by the yield criteria, such as von Mises and Tresca. CONCLUSIONS: Even though differences among the analytical solutions in terms of slump and spread radius existed, the difference can be considered insignificant when the solutions were used as reference to evaluate the appropriateness of numerical approaches, such as the discrete element method.

횡방향 하중을 받는 말뚝의 해석해에 대한 이론적 고찰

이승현 한국방재학회 2011 한국방재학회논문집 Vol.11 No.3

Analytical solutions for laterally loaded piles were derived. Critical pile length which can be considered as the length for behaving as long pile was investigated varying with densities of sandy soils. Lateral behaviors obtained from analytical solution and numerical solution were also investigated. Non-dimensional critical pile lengths obtained from analytical solutions for three types of pile head boundary conditions were 2.33.2. By comparing analytical solutions with numerical solutions, distribution of pile deflection and that of moment were similar and it can be seen that pile head deflection obtained by analytical method is conservative. And the values of moments were not too different between analytical solution and numerical solution. 본 연구에서는 횡방향 하중을 받는 말뚝의 해석 해를 유도하고 그로부터 말뚝의 거동분석과 더불어 긴 말뚝으로의 거동기준이 되는 임계 말뚝길이(critical pile length)를 지반조건을 달리하여 구하고 비교 분석하였다. 또한 해석 해에 의한 말뚝의 거동과 p-y곡선을 적용한 수치해석을 통해 말뚝의 거동을 비교분석하였다. 해석 해에 따르면 밀도를 달리한 지반조건에 대해 무차원 임계 말뚝길이는 해석에서 고려한 세 가지 말뚝머리 경계조건에 있어 2.33.2 사이였다. 해석 해에 의한 결과와 수치해석에 의한 결과를 비교하면 말뚝길이에 따른 말뚝의 변형과 모멘트 분포양상은 유사하였다. 말뚝머리 변형량은 해석에 의한 경우가 수치해석에 의한 경우보다 보수적인 값을 보여주었으며 휨모멘트의 값은 해석에 의한 값과 수치해석에 의한 값 사이에 큰 차이를 보이지 않았다.

축대칭 함몰지형 위를 통과하는 파에 대한 확장형 완경사 방정식의 해석해

정태화(Taehwa Jung),이승오(Seung-Oh Lee),김수영(Sooyoung Kim),류용욱(Yong-Uk Ryu) 한국연안방재학회 2020 한국연안방재학회지 Vol.7 No.2

Many analytical solutions to the extended mild slope equation were presented by adopting Hunt’s approximated solution of linear dispersion in order to investigate how to be affected water wave when wave propagating on a shallow water depth suddenly encounter a much deeper water depth. In this study, an analytical solution to the extended mild slope equation is derived for waves propagating over an axi-symmetric pit in which the water depth varies in proportion to a power of radial distance from the center of it. First, the governing equation is transformed into ordinary differential equation by using the method of separation of variables, and the coefficients expressed in terms of phase velocity and group velocity are transformed into explicit forms by using Hunt’s (1979) approximate solution for wave dispersion. The functions related to higher order terms such as bottom curvature and bottom slope-squared are expressed as power series. Finally, by using the Frobenius series, the analytical solution to the extended mild slope equation is derived. The validity of the analytical solution is demonstrated by comparison with the numerical solution computed by the finite element method. The present solution shows more accurate result than the solution of the mild slope equation while less accurate in intermediate depth region due to the feature of Hunt’s solution.

흐름과 진행파에 의한 해저지반 내 잔류간극수압의 해석해

이광호,김도삼,김동욱,강기천,김태형 한국지반공학회 2015 한국지반공학회논문집 Vol.31 No.7

This study extended the existing analytical solution proposed by Lee et al. (2015a) which has some errors for prediction of the residual pore water pressure into progressive wave and flow coexisting field. At this time, the variation of incident wave period and wave length should be incorporated to Lee et al.’s analytical solution, which does not consider flow. For the case of infinite thickness, the new analytical solution using Fourier series was compared to the analytical solution using Laplace transformation proposed by Jeng and Seymour (2007). It was verified that the new solution was identical to the Jeng and Seymour’s solution. After verification of the new analytical solution, the residual pore water pressure head was examined closely under various given values of flow velocity’s magnitude and direction and incident wave’s period and seabed thickness. In the proposed each analytical solution, it is able to asymptotic approach to shallow depth with the changes in the soil thickness within finite soil thickness, but not to infinite depth. It is also identified that there exists a discrepancy case between the results obtained from the finite and the infinite seabed thicknesses even on the same soil depth.

축대칭 함몰지형 위를 진행하는 파의 변형에 관한 완경사 방정식의 해석 해

정태화(Jung, Tae-Hwa),서경덕(Suh, Kyung-Duck) 한국해안해양공학회 2006 한국해안해양공학회 논문집 Vol.18 No.4

축대칭 함몰지형 위를 진행하는 파에 관한 완경사 방정식의 해석 해를 유도하였다. 함몰지형 내부에서의 수심은 함몰지형 중심으로부터의 거리의 덕에 비례하여 변한다. 변수 분리법을 사용하여 지배 방정식을 상미분 방정식으로 전환하였으며, Hunt(1979)의 근사식을 사용하여 방정식의 계수들을 양함수의 형태로 변환하였다. 마지막으로 Frobenius 급수를 사용하여 해석 해를 유도하였다. Hunt 근사식의 특성 상 본 연구에서 유도된 해석 해는 천해와 심해에서는 정확하고 중간수심에서는 덜 정확하다. 해석 해의 타당성을 검토하기 위하여 수치 모델 결과와 비교하였다. 또한 해석 해를 이용하여 함몰지형의 형상과 상대수심 등을 변화시키면서 다양한 조건에 대한 파랑 변형을 검토하였다. An analytic solution to the mild-slope equation is derived for waves propagating over an axi-symmetric pit. The water depth inside the pit varies in proportion to a power of radial distance from the pit center. The governing equation is transformed into ordinary differential equations by using separation of variables, and the coefficients of the equations are transformed into explicit forms by using Hunt's (1979) approximate solution. Finally, by using the Frobenius series, the analytic solution is derived. Due to the feature of Hunt's equation, the present analytic solution is accurate in shallow and deep waters, while it is less accurate in intermediate depth water. The validity of the analytic solution is demonstrated by comparison with numerical solutions. The analytic solution is also used to examine the effects of pit geometry and relative depth on wave transformation.

구배법 알고리즘에 의한 슬래브축열의 최적제어 해석

정재훈(Jae-Hoon Jung) 대한설비공학회 2008 설비공학 논문집 Vol.20 No.8

In this paper, the optimal bang-bang control problem of an air-conditioning system with slab thermal storage was formulated by gradient method. Furthermore, the numeric solution obtained by gradient method algorithm was compared with the analytic solution obtained on the basis of maximum principle. The control variable is changed uncontinuously at the start time of thermal storage operation in an analytic solution. On the other hand, it is showed as a continuous solution in a numeric solution. The numeric solution reproduces the analytic solution when a tolerance for convergence is applied severely. It is conceivable that gradient method is effective in the analysis of the optimal bang-bang control of the large-scale system like an air-conditioning system with slab thermal storage.

Restrictions on the analytic approach of unconstrained melting inside a spherical capsule

홍상우,이용태,정재동 대한기계학회 2015 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.29 No.11

Analytic solutions are very important for providing physical insight and a basis for the examination of numerical and experimental results. However, analytic approaches are subject to extensive assumptions, so they cannot explain the complex phenomena involved withunconstrained melting. On the discharging conditions in real Thermal energy storage (TES) systems, such as capsule size, wall temperatureand capsule materials, the assumptions used in the analytic solutions of unconstrained melting inside a spherical capsule are closelyexamined by comparing them with numerical solutions. Numerical approaches are more practical to implement all these complex behaviors,thus are used as the basis in the comparison after thorough validation. The restrictions on low Stefan number and low Rayleighnumber of existing analytic solutions are addressed.

Analytical solutions for vibrations of rectangular functionally graded Mindlin plates with vertical cracks

Chiung-Shiann Huang,Yun-En Lu 국제구조공학회 2023 Structural Engineering and Mechanics, An Int'l Jou Vol.86 No.1

Analytical solutions to problems are crucial because they provide high-quality comparison data for assessing the accuracy of numerical solutions. Benchmark analytical solutions for the vibrations of cracked functionally graded material (FGM) plates are not available in the literature because of the high level of complexity of such solutions. On the basis of first-order shear deformation plate theory (FSDT), this study proposes analytical series solutions for the vibrations of FGM rectangular plates with side or internal cracks parallel to an edge of the plates by using Fourier cosine series and the domain decomposition technique. The distributions of FGM properties along the thickness direction are assumed to follow a simple power law. The proposed analytical series solutions are validated by performing comprehensive convergence studies on the vibration frequencies of cracked square plates with various crack lengths and under various boundary condition combinations and by performing comparisons with published results based on various plate theories and the theory of three-dimensional elasticity. The results reveal that the proposed solutions are in excellent agreement with literature results obtained using the Ritz method on the basis of FSDT. The paper also presents tabulations of the first six nondimensional frequencies of cracked rectangular Al/Al2O3 FGM plates with various aspect ratios, thickness-to-width ratios, crack lengths, and FGM power law indices under six boundary condition combinations, the tabulated frequencies can serve as benchmark data for assessing the accuracy of numerical approaches based on FSDT.