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.

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.

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.

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.

1차원 해석해를 이용한 저투수성 매체에서의 확산에 관한 연구

장성간,양민준 대한자원환경지질학회 2020 자원환경지질 Vol.53 No.1

One-dimensional analytical solutions were used for forward and back diffusion of trichloroethylene (TCE) and tetrachloroethylene (PCE) in a single system with high- and low-permeability layers. Concentration profiles in a lowpermeability layer, diffusive fluxes at the interface between the high- and low-permeability layers, and contaminant persistence in the high-permeability layer due to back diffusion were simulated with a comparison of semi-infinite and finite analytical solutions. In order to validate the analytical solutions used in this study, the results of onedimensional analytical solutions developed by Yang et al. (2015) were compared with Nash–Sutcliffe model efficiency coefficient (NSE). When compared with Yang et al. (2015), the analytical solutions used in this study showed good agreements (NSE = 0.99). When compared with semi-infinite analytical solutions, TCE and PCE concentration profiles in the low-permeability layer, the diffusive fluxes, and the contaminant tailings of the high-permeability layer were underestimated. In order to determine the appropriate analytical solutions based on the effective diffusion coefficient, the thickness of the low-permeability layer, and the diffusion time in the TCE and PCE contaminated site, a term of dimensionless diffusion length (Zd) was used. If the Zd is less than 0.7, the semi-infinite solutions can be used to simulate accurate concentration profiles in low-permeability layers. If the Zd is greater than 0.7, the reliability of simulations may be improved by using the finite solutions.

Research on a Key Question in the Parlange Solution

Yuelu Zhu,Yaoting Xiao 대한토목학회 2022 KSCE JOURNAL OF CIVIL ENGINEERING Vol.26 No.1

In order to solve the problem that the key parameter c(t) in the Parlange Solution of Richards Equation is difficult to calculate, this paper proposes a variable separation method based on experimental, which transforms the integral equation containing c(t) into transcendental equation and finds the analytical solution. The calculation results show that when the test parameters D(θ) and k(θ) are power functions, the improved method converges quickly, a stable solution can be obtained by more than ten iterations, and the generated trajectory of soil moisture content conforms to the actual situation. The proposed solution converges faster than the simplified Parlange Solution, and is very close to the Philip Solution, with a difference of less than 5%. The proposed solution has strict analytical steps and clear mathematical significance, which can be regarded as a supplement to the Parlange Model in a key step.

AN APPROXIMATE ANALYTICAL SOLUTION OF A NONLINEAR HYDRO-THERMO COUPLED DIFFUSION EQUATION

Lee, Jeong-woo,Cho, Won-cheol Korea Water Resources Association 2001 Water engineering research Vol.2 No.3

An approximate analytical solution of a nonlinear hydro-thermo coupled diffusion equation is derived using the dimensionless form of the equation and transformation method. To derive an analytical solution, it is drastically assumed that the product of first order derivatives in the non-dimensionalized governing equation has little influence on the solution of heat and moisture behavior problem. The validity of this drastic assumption is demonstrated. Some numerical simulation is performed to investigate the applicability of a derived approximate analytical solution. The results show a good agreement between analytical and numerical solutions. The proposed solution may provide a useful tool in the verification process of the numerical models. Also, the solution can be used for the analysis of one-dimensional coupled heat and moisture movements in unsaturated porous media.

누수대수층 지하수 양수에 관한 Hunt 해석해의 적용성 평가

이정우,정일문 대한토목학회 2020 대한토목학회논문집 Vol.40 No.6

In this study, the applicability of Hunt’s analytical solution for a two-layered leaky aquifer system, which was developed to estimate stream depletion due to the groundwater pumping of the upper shallow aquifer, was evaluated. The 5-year averaged stream depletionswere estimated using Hunt’s analytical solution for various combinations of hydraulic characteristic values such as transmissivity, storage coefficient of the two aquifers, interlayer leakage coefficient, stream-well distance, hydraulic conductivity of the streambed, and stream width. Through comparison with the numerical solution accurately simulated with a MODFLOW groundwater flow model, the analytical solution derived by regarding the stream width as a point was evaluated. It was found that the error in the stream depletion calculated by the analytical solution can be reduced to less than 0.05 when the stream-well distance is greater than the stream width orwhen the stream depletion factor (SDF) is more than about 3,000 days. In addition, when the streambed hydraulic conductivity is less than 1 m/d, the hydraulic diffusion coefficient of the lower aquifer layer is less than 100 m2/d, the hydraulic diffusion coefficient ratio of the upper and lower aquifer layers is 5 or more, and the leakage coefficient between the layers is less than 0.0004 m/d, the overall analytical solutions were overestimated compared with the numerical solutions. 본 연구에서는 상, 하부 2개의 대수층이 연결된 누수대수층에 대해 상부 1층 지하수 양수로 인한 하천수 감소량을 산정하기 위해 개발된 Hunt해석해의 적용성을 평가하였다. 두 대수층의 투수량계수 및 저류계수, 층간 누수계수, 하천-관정 이격거리, 하상의 수리전도도, 하폭 등 다양한 수리특성치 크기 조합에 따라 양수 후 5년 평균 하천수 감소량을 해석해로 산정하고, 범용적으로 사용되고 있는 MODFLOW 지하수유동모형으로 정교하게 모의된 수치해와의 비교를 통해 하폭을 점으로 간주하여 유도한 해석해 적용 결과가 정확한지를 평가하였다. 그 결과 하천-관정 이격거리가 하폭 이상되거나, 하천고갈인자(Stream Depletion Factor)가 약 3,000일 이어야 해석해로 산정한 하천수 감소비의 오차를 0.05 아래로 줄일 수 있는 것으로 분석되었다. 또한 하상수리전도성이 1 m/d 보다 작거나, 하부대수층의 수리확산계수가 100 m2/d 보다 작거나, 상부와 하부대수층의 수리확산계수비가 5 이상, 층간 누수계수가 0.0004 m/d 보다 작을 때 해석해가 수치해에 비해 전반적으로 과다하게 산정되는 것으로 분석되었다.

Evaluation of stream depletion due to nearby groundwater pumping using Baalousha analytical solution

Lee, Jeongwoo,Kim, Nam Won,Chung, Il-Moon,Hong, Sung Hun 한국수자원학회 2018 한국수자원학회논문집 Vol.51 No.2

본 연구에서는 하폭, 하상수리전도도 등의 하천수리특성을 고려할 수 있는 Baalousha (2012)의 해석해를 이용하여 안성천 상류구간 인근에 위치한 관정 17개에 대해 지하수 양수로 인한 하천수 감소량을 산정하고 그 특성을 고찰하였다. 해석해 적용을 위해서 양수시험과 시피지시험을 통해 측정한 대수층과 하상의 수리특성값을 이용하였다. 양수기간 5년 동안 양수량 대비 하천수 감소량은 약 0.23에서 0.89로 관정 위치별로 차이가 크게 나타났으며, 하천고갈인자(Stream Depletion Factor, SDF) 값이 1,000일보다 큰 경우 0.4 미만으로 양수의 영향이 작은 것으로 분석되었다. Baalousha (2012) 해석해 적용 결과를 미소하폭에 대한 Hunt (1999) 해석해 적용 결과와 비교한 결과 연구대상 지역은 상대적으로 간단한 Hunt (1999) 해석해로 지하수 양수 영향을 파악하는데 충분한 것으로 분석되었다. 또한 투수량계수, 저류계수, 하상수리전도도, 하폭, 하천-관정 이격거리, 하폭 등의 수리특성치 조합에 따른 총 3,000가지 조건에 대해 각각의 해석해로 5년 평균 하천수 감소비를 구하여 비교한 결과 하천-관정 이격거리가 하폭 보다 길어야 두 해석해의 차이가 작아 하폭의 영향이 감소하는 것으로 분석되었다. This study was to evaluate the stream depletion due to groundwater pumping from 17 wells near the Anseongcheon upper stream using the Baalousha’s analytical solution (2012) which directly considers stream width and streambed hydraulic conductivity. The input hydraulic values of aquifer and streambed were obtained from the pumping tests and seepage experiments. The estimated streamflow depletion rates divided by pumping rate (dimensionless stream depletion) showed a range from 0.23 to 0.89 for 5 year pumping. In particular, the results revealed that the groundwater pumping has insignificant effects on streamflow when the stream depletion factor (SDF) is higher than 1,000 with values of dimensionless stream depletion lower than 0.4. A more simple Hunt’s solution (1999) also applied to the same wells, and the results showed that the difference between the dimensionless stream depletions calculated by using both solutions could be negligible. From the comparison of the Baalousha’s solution (2012) with the Hunt’s solution (1999) with total 3,000 cases of simulations with combinations of various aquifer and stream properties, the stream-well distance should be more longer than stream width for reducing the discrepancy between both solutions.