All Issue

2026 Vol.27, Issue 10
Three-Dimensional Extension of Terzaghi’s Arching Formula Using the Average Inclination Angle of the Slip Surface and Evaluation of its Effects under Various Excavation Conditions

활동면의 평균 경사각을 이용한 Terzaghi 아칭공식의 3차원 확장 및 다양한 굴착조건에서의 영향 평가

Moorak Son1*
손 무락1*
1Professor, Department of Civil Engineering, Daegu University
*Corresponding Author
1 October 2026. pp. 5-12
PDF XML Citation
Abstract

Terzaghi proposed a two-dimensional arching formula by assuming that vertical slip surfaces form in the ground above a downward-moving trapdoor. However, in actual ground conditions, slip surfaces are often inclined rather than vertical, and their inclination angle may vary with depth. To address this limitation, this study extends Terzaghi’s two-dimensional arching formula to three-dimensional excavation conditions and proposes a modified formula that accounts for the depth-dependent variation of slip surface inclination using an average inclination angle. The inclination angle is assumed to be 90° at the ground surface and to change to a specified value at the excavation level. Using the proposed formula, the effects of excavation width, length, depth, and slip surface inclination angle on vertical stress were analyzed. The results show that the depth-dependent variation of slip surface inclination significantly affects vertical stress, and that this effect depends on the excavation conditions. The proposed average-inclination-angle-based three-dimensional arching approach provides a practical and simplified framework for evaluating arching behavior and vertical stress under three-dimensional excavation conditions.

Keywords
Arching
Vertical stress
Three-dimensional excavation
Depth-dependent slip surface
Average inclination angle
Excavation conditions

Terzaghi 트랩도어의 하향 이동으로 상부 지반에 수직 활동면이 형성된다고 가정하고, 아칭효과를 평가하기 위한 2차원 공식을 제안하였다. 그러나 실제 지반에서 형성되는 활동면은 수직이 아니라 경사진 형태를 보이는 경우가 많으며, 그 경사각도 깊이에 따라 변화할 수 있다. 본 연구에서는 이러한 한계를 보완하기 위해 Terzaghi의 2차원 아칭공식을 3차원 굴착조건으로 확장하고, 깊이에 따라 변화하는 활동면 경사각의 영향을 평균 경사각으로 반영한 수정식을 제안하였다. 활동면 경사각은 지표면에서 90°이고, 굴착면에서는 특정 경사각으로 변화하는 것으로 가정하였다. 제안식을 이용하여 굴착 폭, 굴착 길이, 굴착 깊이 및 활동면 경사각 조건이 수직응력에 미치는 영향을 분석하였다. 그 결과, 깊이에 따른 활동면 경사각 변화는 수직응력 산정에 큰 영향을 미치며, 그 영향은 굴착조건에 따라 달라지는 것으로 나타났다. 본 연구에서 제안한 평균 경사각 기반 3차원 아칭해석 방법은 깊이에 따라 변화하는 활동면 형상을 보다 현실적으로 고려하면서도 계산 과정을 단순화할 수 있는 실용적인 접근법이다. 따라서 본 연구의 결과는 3차원 굴착조건에서의 아칭현상과 수직응력 변화를 이해하고 평가하는 데 유용한 해석적 틀을 제공하며, 지반공학적 설계 및 해석에 실질적인 기초자료로 활용될 수 있을 것으로 판단된다.

키워드
아칭
수직 응력
3차원 굴착
깊이 의존적 활동면
평균 경사각
굴착 조건
References
1

Adachi, T., Kimura, M., Kishida, K., Kosaka, K. and Sakayama, Y. (1999), The mechanical behavior of tunnel interaction through three dimensional trapdoor tests (in Japanese), J. Geotech. Eng., JSCE 638/III-49, pp. 285~299.

10.2208/jscej.1999.638_285
2

Bierbaumer, A. (1913), Die Dimensionierung des Tunnelmauerwerkes, Leipzig, W. Engelmann.

3

Cain, W. (1916), Earth Pressure, Retaining Walls and Bins, New York, John Wiley & Sons, Inc.

4

Caquot, A. (1934), Equilibre des Massifs Ii Frottement Interne, Paris, GauthierVillard.

5

Chevalier, B., Combe, G. and Villard, P. (2012), Experimental and discrete element modeling studies of the trapdoor problem: influence of the macro-mechanical frictional parameters, Acta Geotechnica, Vol. 7, No. 1, pp. 15~39.

10.1007/s11440-011-0152-5
6

Chevalier, B. and Otani, J. (2010), 3D Arching Effect in the Trap-Door Problem: A Comparison between X-Ray CT Scanning and DEM Analysis, Proc., GeoFlorida 2010, pp. 570~579.

10.1061/41095(365)54
7

Engesser, F. (1882), Über den Erddruck gegen innere Stützwände, Deutsche Bauzeitung, Vol. 16, pp. 91~93.

8

Getzler, Z., Gellert, M. and Eitan, R. (1970), Analysis of Arching Pressures in Ideal Elastic Soil, J. of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM4, pp. 1357~1372.

10.1061/JSFEAQ.0001443
9

Iglesia, G.R., Einstein, H.H. and Whitman, R.V. (2014), Investigation of Soil Arching with Centrifuge Tests, J. Geotech. Geoenviron. Eng., Vol. 140, No. 2, pp. 1~13.

10.1061/(ASCE)GT.1943-5606.0000998
10

Kim, S. Y., Kang, S., Park, G., Lee, D., Lim, Y. and Lee, J. S. (2024), Detection of Roadbed Layers in Mountainous Area Using Down-Up-Crosshole Penetrometer and Ground Penetrating Radar, Measurement, Vol. 224, Article 113889. https://doi.org/10.1016/j.measurement.2023.113889

10.1016/j.measurement.2023.113889
11

Koutsabeloulis, N.C. and Griffiths, D.V. (1989), Numerical Modeling of the Trapdoor Problem, Geotechnique, Vol. 39, No. 1, pp. 77~89.

10.1680/geot.1989.39.1.77
12

Li, L, Aubertin, M. and Belem, T. (2005), Formulation of a three dimensional analytical solution to evaluate stresses in backfilled vertical narrow openings, Canadian Geotechnical Journal, Vol. 42, No. 6, pp. 1705~1717.

10.1139/t05-084
13

Li, L, Dube, J. and Aubertin, M. (2013), An Extension of Marston’s Solution for the Stresses in Backfilled Trenches with Inclined Walls, Canadian Geotechnical Journal, Vol. 42, No. 6, pp. 1705~1717.

14

Marston, A. (1930), The theory of external loads on closed conduits in the light of the latest experiments, Bulletin 96, Iowa State University Engineering Experiment Station, Ames, Iowa.

15

McNulty, J.W. (1965), An Experimental Study of Arching in Sand, Ph.D. Thesis in Civil Engineering, University of Illinois.

16

Nielson, F.D. (1966), Soil structure arching analysis of buried flexible structures, PhD Thesis, University of Arizona, faculty of Civil Engineering.

17

Paikowsky, S.G., DiRocco, K.J. and Xi, F. (1993), Interparticle contact force analysis and measurements using photoelastic techniques, 2nd International Conference on Discrete Element Methods, MIT, Cambridge, Massachusetts, pp. 449~461.

18

Pardo, G.S. and Sáez, E. (2014), Experimental and numerical study of arching soil effect in coarse sand, Computers and Geotechnics, Vol. 57, pp. 75~84.

10.1016/j.compgeo.2014.01.005
19

Pirapakaran, K. and Sivakugan, N. (2007), Arching within hydraulic fill stopes, Getech. Gelo. Eng., Vol. 25, pp. 25~35.

10.1007/s10706-006-0003-6
20

Singh, S., Shukla, S.K. and Sivakugan, N. (2011), Arching in inclined and vertical mine stopes, Geotech. Geol. Eng., Vol. 29, pp. 685~693.

10.1007/s10706-011-9410-4
21

Sivakugan, N., Widisinghe, S. and Wang, V.Z. (2014), Vertical stress determination within backfilled mine stopes, Int. J. of Geo. Mech., Vol. 14, No. 5, pp. 1~5.

10.1061/(ASCE)GM.1943-5622.0000367
22

Son, M. (2017), Three-dimensional expansion of the Terzaghi arching formula considering inclined sliding surfaces and examination of its effects, Int. J. of Geo. Mech., Vol. 17, No. 7, 06016043.

10.1061/(ASCE)GM.1943-5622.0000844
23

Spangler, M.G. and Handy, R.L. (1982), Loads on Underground Conduits, Soil Engineering, New York, pp. 727~763.

24

Terzaghi, K. (1943), Theoretical soil mechanics, John Wiley and Sons, New York.

10.1002/9780470172766
25

Vardoulakis, I., Graf, B. and Gudehus, G. (1981), Trap-door problem with dry sand: A statical approach based upon model test kinematics, International journal for numerical and analytical methods in geomechanis, Vol. 5, No. 1, pp. 57~78.

10.1002/nag.1610050106
26

Völlmy, A. (1937), Eingebettete Rohre. Mitt. Inst. Baustatik, Eidgen. Tech. Hochschule, Zurich, Mitt. No.9.

Information
  • Publisher :Korean Geo-Environmental Society
  • Publisher(Ko) :한국지반환경공학회
  • Journal Title :Journal of the Korean Geo-Environmental Society
  • Journal Title(Ko) :한국지반환경공학회 논문집
  • Volume : 27
  • No :10
  • Pages :5-12
  • Received Date : 2026-05-30
  • Revised Date : 2026-06-09
  • Accepted Date : 2026-09-18
  • DOI :https://doi.org/10.14481/jkges.2026.27.10.5
Journal Informaiton Journal of the Korean Geo-Environmental Society Journal of the Korean Geo-Environmental Society
Online Submission
  • NRF
  • KOFST
  • KISTI Current Status
  • KISTI Cited-by
  • orcid
Journal Informaiton Journal Informaiton - close