Linear Strain Triangle Mathematics: Stiffness, Stress and Consistent Load Vector


  • Mohamed Abdalla Almheriegh Associate Professor, Department of civil Engineering,Faculty of Engineering - Tripoli University,P O Box 82677, Tripoli, Libya


consistent load vector for conical axe-symmetric shell element or pressure vessel element, modeling of multiple materials, mathematics of finite element LST, stiffness matrix, stress and, consistent load vector


Typically, linear strain triangle LST is widely used for elastic analysis not only due to its ease but also due to the good results it can lead to over the traditional constant strain triangle CST. In plane elastic analysis of two combined materials the LST is quite reliable and trust worthy in terms of stress results for the same element size, LST is very useful in modelling combined materials thus believed to be efficient and can easily take into consideration, self-weight of used materials, strains due to causes other than loading; such as moisture, temperature, creep and shrinkage are easily incorporated. A mathematical formulation of stiffness matrix, stress and the more rarely dealt with consistent load vector for loads distributed on element edge are proved and highlighted for young engineers who mostly dealing with readily used FE codes, the formula proved for LST element load vector is further extended and thus a consisted load vector for conical axe-symmetric shell element is introduced.


Desai, C. S., and J. F. Adel, Introduction to the Finite Element Method, (New York: Van Nostrand Reinhold, 1972)

Zeinkiewicz. O. C.M, The Finite Element Method in Engineering Science, Second edition (New York: McGraw Hill, 1971)

William H. Bowes and Leslie T. Russel, Stress Analysis by the Finite Element Method for practicing Engineers, Lixengton Books. D.C. Heath and Company, Toronto, 1975

Mohamed Almheriegh. “Initial Strain Problems Formulation of Combined Materials” IOSR Journals of Mechanical and Civil Engineering, IOSR Journals, International Organization of Scientific Research,. Volume: 11 issue: 1, pp 37-47 (Version -4) e-ISSN: 2278-1684 p-ISSN: 2320-234X.

. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer Science + Buiness Media LLC. 3rd eds., 2008.

. McNeal R. H., (1978) A Simple Quadrilateral Shell Element, Computers and Structures, Vol. 8, pp. 175-183.

. Clough R. W. and Tocher J. L, (1965) Finite Element Stiffness Matrices for Analysis of Plate Bending, Proc. Conference on Matrix Methods in Structural Mechanic, WPAFB, Ohio, pp. 515-545.

. Green B. E., Strome D. R., and Weikel R. C.,( 1961) Application of the stiffness method to the analysis of shell structures, Procedures on Aviation Conference, American Society of Mechanical Engineers, Los Angeles, arch.

. Bazeley, G. P., Cheung Y. K., Irons B. M. and Zienkiewicz, O. C., (1966) Triangular Elements in Plate Bending, Confirming and Non – Confirming Solutions, Proc. 1st Conference on Matrix Methods in Structural Mechanics, pp. 547-576, Wright Patterson AF Base, Ohio.

Cleveland State University, FE course

. Cowper, G.R.: Gaussian quadrature formulas for triangles. Int. J. Numer. Methods Eng. 7, 405–408 (1973)

. Flores, F.G.: A two-dimensional linear assumed strain triangular element for finite deformation analysis. J. Appl. Mech. 73, 970–976 (2006)

. Cook, R.D.: On the Allman triangle and related quadrilateral element. Comput. Struct. 22, 1065–1067 (1986)




How to Cite

Mohamed Abdalla Almheriegh. (2021). Linear Strain Triangle Mathematics: Stiffness, Stress and Consistent Load Vector. American Scientific Research Journal for Engineering, Technology, and Sciences, 83(1), 1–11. Retrieved from