Skip to main content
SpringerLink
Log in

Fundamentals of the Finite Element Method

  • Chapter
  • First Online:
Computational Geomechanics and Hydraulic Structures

Part of the book series: Springer Tracts in Civil Engineering ((SPRTRCIENG))

  • 825 Accesses

Abstract

This chapter summarizes the principles, governing equations, and basic algorithms related to the finite element method (FEM), which has been the most general and powerful computation tool in the engineering design and analysis since the later 1960s. The “standard” and “hierarchical” shape functions using orthogonal polynomial series are elucidated in details, based on which the algorithms with regard to the fields of permeability, temperature, and deformation are elaborated. The solution techniques particularly important for hydraulic structures related to the issues of rock EDZ, phreatic surface, concrete hydration heating, reservoir water temperature, dynamic response, are addressed. This chapter is closed with the discussion on the safety criteria of hydraulic structures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Ang AH-S, Tang WH. Probability concepts in engineering planning and design, vol. I & II. New York: Wiley; 1984.

    Google Scholar 

  • Argyris JH, Kelsey S. Energy theorems and structural analysis. London: Butterworth Scientific Publications; 1960.

    Book  MATH  Google Scholar 

  • Babuška I, Szabó BA, Katz IN. The p-version of the finite element method. SIAM J Numer Anal. 1981;18(3):515–45.

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška I, Griebel M, Pitkäranta J. The problem of selecting the shape functions for a p-type finite element. Int J Numer Meth Eng. 1989;28(8):1891–908.

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška I, Henshaw WD, Oliger JE, Flaherty JE, Hopcroft JE, Tezduyar T. Modeling, mesh generation, and adaptive numerical methods for partial differential equations. New York: Springer; 1995.

    Book  MATH  Google Scholar 

  • Bathe KJ, Khoshgoftaar MR. Finite element free surface seepage analysis without mesh iteration. Int J Numer Anal Meth Geomech. 1979;8(3):13–22.

    Article  MATH  Google Scholar 

  • Bathe KJ, Wilson EL. Stability and accuracy analysis of direct integration methods. Earthquake Eng Struct Dynam. 1972;1(3):283–91.

    Article  Google Scholar 

  • Bednarz TE, Lei CW, Patterson JC. Unsteady natural convection induced by diurnal temperature changes in a reservoir with slowly varying bottom topography. Int J Therm Sci. 2009;48(10):1932–42.

    Article  Google Scholar 

  • Brown PT, Booker JR. Finite element analysis of excavation. Comput Geotech. 1985;1(3):207–20.

    Article  Google Scholar 

  • Cai M, Kaiser PK. Assessment of excavation damaged zone using a micromechanics model. Tunn Undergr Space Technol. 2005;20(4):301–10.

    Article  Google Scholar 

  • Caughey TH, Kelly MEJ. Classical normal modes in damped linear dynamic systems. J Appl Mech ASME. 1965;32(3):583–8.

    Article  MathSciNet  Google Scholar 

  • Chen SH. Hydraulic structures. Berlin: Springer; 2015.

    Book  Google Scholar 

  • Chen SH, Wang WM, She CX, Xu MY. Unconfined seepage analysis of discontinuous rock slope. J Hydrodyn. 2000;12(3):75–86.

    MATH  Google Scholar 

  • Chen SH, Chen SF, Shahrour I, Egger P. The feedback analysis of excavated rock slope. Rock Mech Rock Eng. 2001;34(1):39–56.

    Article  Google Scholar 

  • Chen SH, Qin WX, Shahrour I. Comparative study of sock slope stability analysis methods for hydropower projects. Mech Res Commun. 2007;34(1):63–8.

    Article  Google Scholar 

  • Chen SH, Zhang GX, Zhu YM. Thermal stresses and temperature control of concrete. In: Zhou JP, Dang LC, editors. Handbook of hydraulic structure design, vol. 5. Concrete dams (Chapter 6). Beijing: China Water Power Press; 2011a (in Chinese).

    Google Scholar 

  • Chen YF, Hu R, Zhou CB, Li DQ, Rong G. A new parabolic variational inequality formulation of Signorini’s condition for non-steady seepage problems with complex seepage control systems. Int J Num Anal Meth Geomech. 2011b;35(9):1034–58.

    Article  MATH  Google Scholar 

  • Chen SH, Wang GJ, Zhou H, Wang WM, Zou LC. Evaluation of excavation-induced relaxation and its application to an arch dam foundation. Int J Numer Anal Meth Geomech. 2012;36(1):166–81.

    Article  Google Scholar 

  • Chen GQ, Huang RQ, Zhou H, et al. Research on progressive failure for slope using dynamic strength reduction method. Rock Soil Mech. 2013;34(4):1140–6 (in Chinese).

    Google Scholar 

  • Chen HQ, Wu SX, Dang FN. Seismic safety of high arch dams. Beijing: Academic Press; 2015.

    Google Scholar 

  • Cheng Z, Chen SH. Three dimensional hierarchical finite element method for hydraulic structure. J Hydraul Eng. 1999;21(12):53–8 (in Chinese).

    Google Scholar 

  • Chilton L, Suri M. On the selection of a locking-free hp element for elasticity problems. Int J Numer Meth Eng. 1997;40(11):2045–62.

    Article  MathSciNet  MATH  Google Scholar 

  • Chopra AK. Simplified earthquake of concrete gravity dam. J Struct Div ASCE. 1987;113(ST8):1688–708.

    Google Scholar 

  • Chopra AK, Chakrabarti P. Earthquake analysis of concrete gravity dams including dam-water-foundation rock interaction. Earthquake Eng Struct Dynam. 1981;9(4):363–83.

    Article  Google Scholar 

  • Chung KY, Kikuchi N. Adaptive methods to solve free boundary problems of flow through porous media. Int J Num Anal Meth Geomech. 1987;11(1):17–31.

    Article  MATH  Google Scholar 

  • Clough RW. The finite element in plane stress analysis. J Struct Div ASCE (Proceedings of the 2nd ASCE conference on electronic computation). 1960;23:345–78.

    Google Scholar 

  • Clough RW, Penzien J. Dynamics of structures, 3rd ed. Berkeley: Computers & Structures Inc; 2003.

    Google Scholar 

  • Cornell CA. A probability based structural code. ACI J. 1969;66(12):974–85.

    Google Scholar 

  • Corthésy R, Leite MH. A strain-softening numerical model of core discing and damage. Int J Rock Mech Min Sci. 2008;45(3):329–50.

    Article  Google Scholar 

  • Courant R. Variational methods for the solution of problems of equilibrium and vibration. Bull Am Math Soc. 1943;49(1):1–61.

    Article  MathSciNet  MATH  Google Scholar 

  • Crisfield MA. Non-linear finite element analysis of solid and structures, vol. 2. New York: Wiley; 1997.

    Google Scholar 

  • Dawson EM, Roth WH, Drescher A. Slope stability analysis by strength reduction. Géotechnique. 1999;49(6):835–40.

    Article  Google Scholar 

  • Desai CS. Finite element residual schemes for unconfined flow. Int J Numer Meth Eng. 1976;10(6):1415–8.

    Article  MATH  Google Scholar 

  • Desai CS, Drumm EC, Zaman MM. Thin layer element for interfaces and joints. J Geotech Eng ASCE. 1985;111(6):793–815.

    Article  Google Scholar 

  • Du L, Du L, Peng S, Wang Y. Back calculations of formation elastic properties in VTI media. World Geol. 2001;20(4):396–416.

    Google Scholar 

  • Duhamel JMC. Mémoir sur la méthode générale relative au movement de le chaleur dans les corps solides plongers dans des milieux dont la temperature varie avec le temps. J de L’ École Polytechnique Paris. 1833;14:20–77.

    Google Scholar 

  • Duncan JM. State of the art: limit equilibrium and finite element analysis of slopes. J Geotech Geoenviron Eng ASCE. 1996;122(7):577–96.

    Article  Google Scholar 

  • Ellingwood B, MacGregor JG, Galambos TV, Cornell CA. Probability based load criteria: load factors and load combinations. J Struct Div ASCE. 1982;108(ST5):978–97.

    Google Scholar 

  • Exadaktylos G, Tsoutrelis CE. Pillar failure by axial splitting in brittle rocks. Int J Rock Mech Min Sci Geomech Abstr. 1995;32(6):551–62.

    Article  Google Scholar 

  • Exadaktylos G, Tsouvala S, Liolios P, Barakos G. A three-dimensional model of an underground excavation and comparison with in situ measurements. Int J Num Anal Meth Geomech. 2007;31(3):411–33.

    Article  Google Scholar 

  • Fanelli DIM, Giuseppetti G. Techniques to evaluate effects of internal temperatures in mass concrete. Water Power Dam Constr. 1975;27(7):226–30.

    Google Scholar 

  • Farias MM, Naylor DJ. Safety analysis using finite element. Comput Geotech. 1988;22(2):165–81.

    Article  Google Scholar 

  • FEMA. Federal guidelines for dam safety: earthquake analyses and design of dams. Washington, DC: FEMA; 2005.

    Google Scholar 

  • Fenton GA, Griffiths DV. A mesh deformation algorithm for free surface problems. Int J Num Anal Meth Geomech. 1997;21(12):817–24.

    Article  Google Scholar 

  • France PW, Parekh CJ, Peters JC. Numerical analysis of free surface seepage problems. J Irrig Drainage Div ASCE. 1971;97(1):165–79.

    Google Scholar 

  • Fu CH, Chen SH. Study on instability criteria of surrounding rock of underground engineering cavern based on catastrophe theory. Rock Soil Mech. 2008;29(1):167–72 (in Chinese).

    Google Scholar 

  • Fujii Y, Kiyama T, Ishijima Y, Kodama J. Examination of a rock failure criterion based on circumferential tensile strain. Pure Appl Geophys. 1998;152(3):551–77.

    Article  Google Scholar 

  • Golshani A, Okui Y, Oda M, Takemura T. A micromechanical model for brittle failure of rock and its relation to crack growth observed in triaxial compression tests of granite. Mech Mater. 2006;38(4):287–303.

    Article  Google Scholar 

  • Goodman RE, Taylor R, Brekke TL. A model for the mechanics of jointed rock. J Soil Mech Found Div ASCE. 1968;94(SM3):637–60.

    Google Scholar 

  • Griffiths DV, Lane PA. Slope stability analysis by finite element. Géotechnique. 1999;49(3):387–403.

    Article  Google Scholar 

  • Hasofer AM, Lind NC. An exact and invariant first order reliability format. J Eng Mech Div ASCE. 1974;100(EMl):111–21.

    Google Scholar 

  • Hinton E, Irons B. Least squares smoothing of experimental data using finite elements. Strain. 1968;4(3):24–7.

    Article  Google Scholar 

  • Howarth DF. Apparatus to determine static and dynamic elastic moduli. Rock Mech Rock Eng. 1984;17(4):255–64.

    Article  Google Scholar 

  • Hrennikoff A. Solution of problems of elasticity by the frame-work method. ASME J Appl Mech. 1941;8(1):A619–715.

    MathSciNet  MATH  Google Scholar 

  • Huber WC, Harleman DRF, Ryan PJ. Temperature prediction in stratified reservoirs. J Hydraul Div ASCE. 1972;98(4):645–66.

    Google Scholar 

  • ICOLD. Computational procedures for dam engineering—reliability and applicability (Bulletin 122). Paris: ICOLD; 2001.

    Google Scholar 

  • ICOLD. Seismic design and evaluation of structures appurtenant to dams (Bulletin 123). Paris: ICOLD; 2002.

    Google Scholar 

  • Imberger J, Patterson J, Hebbert B, Loh I. Dynamics of reservoir of medium size. J Hydraul Div ASCE. 1978;104(HY5):725–43.

    Google Scholar 

  • Jäger P, Schmalholz SM, Schmid DW, Kuhl E. Brittle fracture during folding of rocks: a finite element study. Philos Mag. 2008;88(28–29):3245–63.

    Article  Google Scholar 

  • Kaiser PK, Amann F, Steiner W. How highly stressed brittle rock failure impacts tunnel design. In: Zhao J, et al., editors. Rock mechanics in civil and environmental engineering. Lausanne: ISRM; 2010. p. 27–38.

    Chapter  Google Scholar 

  • Kim JK, Lee SR. An improved search strategy for the critical slip surface using finite element stress fields. Comput Geotech. 1997;21(4):295–313.

    Article  Google Scholar 

  • Kwaśniewski M. Mechanical behavior of anisotropic rocks. Compr Rock Mech. 1993;4(1):16–22.

    Google Scholar 

  • Kwaśniewski M, Takahashi M. Strain-based failure criteria for rocks: state of the art and recent advances. In: Zhao J, et al., editors. Rock mechanics in civil and environmental engineering. Lausanne: ISRM; 2010. p. 45–56.

    Chapter  Google Scholar 

  • Lacy LL. Dynamic rock mechanics testing for optimized fracture designs. In: SPE annual technical conference and exhibitions. San Antonio, TX: Society of Petroleum Engineers; 1997. p. 23–36 (Paper SPE38716).

    Google Scholar 

  • Lechman JB, Griffiths DV. Analysis of the progression of failure of earth slopes by finite elements. In: Griffiths DV, et al., editors. Geotechnical special publication: slope stability. Denver, Colorado: ASCE; 2000. p. 250–65.

    Google Scholar 

  • Leger P, Leclerc M. Hydrostatic, temperature, time-displacement model for concrete dams. J Eng Mech ASCE. 2007;133(3):267–77.

    Article  Google Scholar 

  • Lei CW, Patterson JC. Natural convection in a reservoir sidearm subject to solar radiation: a two-dimensional simulation. Numer Heat Transfer Part A. 2002;42(1–2):13–32.

    Article  Google Scholar 

  • Li X, Wu Z, Takahashi M, Yasuhara K. An experimental study on the failure strain-based failure criteria of brittle materials. J Appl Mech. 2000;3(8):387–94.

    Article  Google Scholar 

  • Liu XR, Tu YL, Zhong ZL, Liu YQ. Slope’s failure criterion based on energy catastrophe in shear strength reduction method. J Central South Univ (Sci Technol). 2016;47(6):2065–72 (in Chinese).

    Google Scholar 

  • Luan MT, Wu YJ, Nian TK. A Criterion for evaluating slope stability based on development of plastic zone by shear strength reduction FEM. J Disaster Prev Mitig Eng. 2003;23(3):1–8.

    Google Scholar 

  • Madsen HO, Krenk S, Lind NC. Methods of structural safety. Englewood Cliffs: Prentice Hall; 1986.

    Google Scholar 

  • Maejima T, Morioka H, Mori T, Aoki K. Evaluation of loosened zones on excavation of a large underground rock cavern and application of observational construction techniques. Tunn Undergr Space Technol. 2003;18(2):223–32.

    Article  Google Scholar 

  • Mahtab M, Goodman RE. Three dimensional analysis of joint rock slope. In: Proceedings of the 2nd ISRM congress, vol. 3. Beograd: Privredni Pregled; 1970, p. 353–60.

    Google Scholar 

  • Malmgren L, Saiang D, Töyrä J, Bodare A. The excavation disturbed zone (EDZ) at Kiirunavaara mine, Sweden—by seismic measurements. J Appl Geophys. 2007;61(1):1–15.

    Article  Google Scholar 

  • Manzari MT, Nour MA. Significance of soil dilatancy in slope stability analysis. J Geotech Geoenviron Eng ASCE. 2000;123(1):75–80.

    Article  Google Scholar 

  • Martino JB, Chandler NA. Excavation-induced damage studies at the Underground Research Laboratory. Int J Rock Mech Min Sci. 2004;41(8):1413–26.

    Article  Google Scholar 

  • Matsui T, San KC. Finite element slope stability analysis by shear strength reduction technique. Soils Found. 1992;32(1):59–70.

    Article  Google Scholar 

  • Mehta PK, Monteiro PJM. Concrete: microstructure, properties, and materials, 3rd ed. New York: McGraw-Hill; 2006.

    Google Scholar 

  • Moura A, Lei XL, Nishisawa O. Prediction scheme for the catastrophic failure of highly loaded brittle materials or rocks. J Mech Phys Solids. 2005;53(11):2435–55.

    Article  MATH  Google Scholar 

  • Nasser KW, Neville AM. Creep of concrete at elevated temperatures. ACI J. 1965;62(12):1567–79.

    Google Scholar 

  • Neumann ST. Saturated-unsaturated seepage by finite element. J Hydraul Div ASCE. 1973;99(12):2233–50.

    Google Scholar 

  • Neumann ST, Witherspoon PA. Finite element method of analyzing of steady seepage with a free surface. Water Resour Res. 1970;6(3):889–97.

    Article  Google Scholar 

  • Neville AM, Diger WH, Brooks JJ. Creep of plain and structural concrete. London: Construction Press; 1983.

    Google Scholar 

  • Newmark NM. A method of computation for structural dynamics. J Eng Mech Div ASCE. 1959;85(EM3):67–94.

    Google Scholar 

  • Oden JT. Finite elements of nonlinear continua. New York: Dover Publications; 2006.

    MATH  Google Scholar 

  • Orlob GT, Selna IG. Temperature variation in deep reservoirs. J Hydraul Div ASCE. 1983;108(HY3):301–25.

    Google Scholar 

  • Owen DRJ, Hinton E. Finite elements in plasticity: theory and practice. Swansea: Pineridge Press Ltd; 1980.

    MATH  Google Scholar 

  • Owen DRJ, Hinton E, Onate E. Computational plasticity: models, software and applications. Swansea: Pineridge Press Ltd; 1989.

    MATH  Google Scholar 

  • Paulay T, Priestley MJN. Seismic design of reinforced concrete and masonry buildings. New York: Wiley; 1992.

    Book  Google Scholar 

  • Ramamurth T. Strength and modulus response of anisotropic rocks. Compr Rock Mech. 1993;4(1):23–8.

    Google Scholar 

  • Raphael JM. Prediction of temperatures in rivers and reservoirs. J Power Div ASCE. 1962;88(2):157–81.

    Google Scholar 

  • Sakurai S. Direct strain evaluation technique in construction of underground opening. In: Proceedings of the 22nd US symposium on rock mechanics. Cambridge: MIT; 1981, p. 278–82.

    Google Scholar 

  • Sakurai S, Kawashima I, Otani T. A criterion for assessing the stability of tunnels. In: Ribeiro e Sousa L, Grossmann NF, editors. Proceedings of the 1993 ISRM international symposium EUROCK 93. Rotterdam: AA Balkema; 1995, p. 969–73.

    Google Scholar 

  • Severn RT. The aseismic design of concrete dams (part one). Int J Water Power Dam Constr. 1978;28(1):37–8 (Part one), 28(2):41–6 (Part two).

    Google Scholar 

  • Singh M. State-of-the-art finite element computer programs for thermal analysis of mass concrete structures. Civil Eng Practicing Des Eng. 1985;4(1):129–36.

    Google Scholar 

  • Sloan SW. Geotechnical stability analysis. Géotechnique. 2013;63(7):531–72.

    Article  Google Scholar 

  • Snitbhan N, Chen WF. Elasto-plastic large deformation analysis of soil slopes. Comput Struct. 1978;9(6):567–77.

    Article  MATH  Google Scholar 

  • Stacey TR. A simple extension strain criterion for fracture of brittle rock. Int J Rock Mech Min Sci Geomech Abstr. 1981;18(6):469–74.

    Article  Google Scholar 

  • Stefan HG, Ford DE. Temperature dynamics in dimictic lakes. J Hydraul Div ASCE. 1975;101(1):97–114.

    Google Scholar 

  • Tatro SB, Schrader EK. Thermal considerations for roller-compacted concrete. ACI J. 1985;82(2):119–28.

    Google Scholar 

  • Taylor RL, Brown CB. Darcy flow solutions with a free surface. J Hydraul Div ASCE. 1967;93(HY2):25–33.

    Google Scholar 

  • Troxell GE, Raphael JM, Davis RE. Long-time creep and shrinkage tests of plain and reinforced concrete. Proc ASTM. 1958;58(3):1101–20.

    Google Scholar 

  • Turner MJ, Clough RW, Martin HC, Topp LJ. Stiffness and deflection analysis of complex structures. J Aeronaut Sci. 1956;23(9):805–23.

    Article  MATH  Google Scholar 

  • Ugai K, Leschinsky D. Three dimensional limit equilibrium and finite element analyses: a comparison of results. Soil Found. 1995;29(4):1–7.

    Article  Google Scholar 

  • Webb JP, Abouchacra R. Hierarchal triangular elements using orthogonal polynomials. Int J Numer Meth Eng. 1995;38(2):245–57.

    Article  MathSciNet  MATH  Google Scholar 

  • Westergaard HM. Water pressure on dams during earthquakes. Trans ASCE. 1933;98(2):418–33.

    Google Scholar 

  • Wilson EL. Three dimensional static and dynamic analysis of structures: a physical approach with emphasis on earthquake engineering, 2nd ed. Berkeley, CA: Computers and Structures Inc; 1998.

    Google Scholar 

  • Wilson EL, Button MR. Three-dimensional dynamic analysis for multi-component earthquake spectra. Earthquake Eng Struct Dyn. 1982;10(3):471–6.

    Article  Google Scholar 

  • Wilson EL, Khalvati M. Finite elements for the dynamic analysis of fluid-solid systems. Int J Numer Meth Eng. 1983;19(11):1657–68.

    Article  MATH  Google Scholar 

  • Wright G, Ulhawwy H, Duncan JM. Accuracy of equilibrium slope stability analysis. J Soil Mech ASCE. 1973;99(SN10):783–91.

    Google Scholar 

  • Yamagami T, Ueta Y. Search for critical slip lines in finite element stress fields by dynamic programming. In: Proceedings of the 6th international conference on numerical methods in geomechanics. Rotterdam: AA Balkema; 1988, p. 1335–9.

    Google Scholar 

  • Yosida Y. Functional analysis, 6th ed. Berlin: Springer; 1980.

    Google Scholar 

  • Zeeman EC. Catastrophe theory: selected papers 1972–1977. Massachusetts: Addison Wesley Publishing Co; 1977.

    MATH  Google Scholar 

  • Zhang YL, Baptista AM. SELFE: a semi-implicit Eulerian-Lagrangian finite element model for cross-scale ocean circulation. Ocean Model. 2008;21(3–4):71–96.

    Article  Google Scholar 

  • Zhang YT, Chen P, Wang L. Residual flow method for seepage problem with free surface. J Hydraul Eng. 1988;10(8):18–26 (in Chinese).

    Google Scholar 

  • Zheng H, Liu DF, Li CG. On the assessment of failure in slope stability analysis by the finite element method. Rock Mech Rock Eng. 2008;41(4):629–39.

    Article  Google Scholar 

  • Zhou HM, Sheng Q, Li WS. Excavation-disturbed zone and weaken degree of mechanical properties for rock mass of TGP shiplock slope. Chin J Rock Mech Eng. 2004;23(7):1078–81 (in Chinese).

    Google Scholar 

  • Zhu BF. Finite element method—principle and application, 2nd ed. Beijing: China Water Power Press; 1998 (in Chinese).

    Google Scholar 

  • Zhu BF. Thermal stresses and temperature control of mass concrete. Beijing: China Electric Power Press; 1999 (in Chinese).

    Google Scholar 

  • Zienkiewicz OC, Cheung YK. The finite element method for analysis of elastic isotropic and orthotropic slabs. Proc Inst Civil Eng. 1964;28(4):471–88.

    Google Scholar 

  • Zienkiewicz OC, Taylor RL. The finite element method for solid and structural mechanics. New York: McGraw Hill; 1967.

    MATH  Google Scholar 

  • Zienkiewicz OC, Mayer P, Cheung YK. Solution of anisotropic seepage by finite elements. J Eng Mech Div ASCE. 1966;92(1):111–20.

    Google Scholar 

  • Zienkiewicz OC, Irons BM, Scott IC, Cambell JS. Three-dimensional stress analysis. In: Proceedings of the IUTAM symposium on high speed computing of elastic structures. Liege: University of Liege Press; 1970, p. 413–33.

    Google Scholar 

  • Zienkiewicz OC, Taylor RL, Zhu JZ. The finite element method—its basis & fundamentals, 6th ed. Oxford: Elsevier Butterworth-Heinemann; 2005.

    Google Scholar 

  • Zienkiewicz OC, Taylor RL, Fox DD. The finite element method for solid and structural mechanics, 7th ed. Oxford: Butterworth-Heinemann; 2013.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Water Resources and Hydropower Engineering, Wuhan University, Wuhan, Hubei, P.R. China

    Sheng-Hong Chen

Authors
  1. Sheng-Hong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sheng-Hong Chen .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Chen, SH. (2019). Fundamentals of the Finite Element Method. In: Computational Geomechanics and Hydraulic Structures. Springer Tracts in Civil Engineering . Springer, Singapore. https://doi.org/10.1007/978-981-10-8135-4_4

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-8135-4_4

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-8134-7

  • Online ISBN: 978-981-10-8135-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation