A BEM solution to transverse shear loading of beams

In this paper a boundary element method is developed for the solution of the general transverse shear loading problem of beams of arbitrary simply or multiply connected constant cross section. The analysis of the beam is accomplished with respect to a coordinate system that has its origin at the centroid of the cross section, while its axes are not necessarily the principal ones. The transverse shear loading is applied at the shear center of the cross section, avoiding in this way the induction of a twisting moment. Two boundary value problems that take into account the effect of Poisson’s ratio are formulated with respect to stress functions and solved employing a pure BEM approach, that is only boundary discretization is used. The evaluation of the transverse shear stresses at any interior point is accomplished by direct differentiation of these stress functions, while both the coordinates of the shear center and the shear deformation coefficients are obtained from these functions using only boundary integration. Numerical examples with great practical interest are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method. The accuracy of the proposed shear deformation coefficients compared with those obtained from a 3-D FEM solution of the ‘exact’ elastic beam theory is remarkable.

[1]  Leonard R. Herrmann,et al.  Elastic Shear Analysis of General Prismatic Beams , 1968 .

[2]  Göran Tolf.  Saint-Venant bending of an orthotropic beam , 1985 .

[3]  Walter D. Pilkey.  Analysis and design of elastic beams , 2002 .

[4]  N. G. Stephen.  Timoshenko’s Shear Coefficient From a Beam Subjected to Gravity Loading , 1980 .

[5]  Pierre Ladevèze,et al.  New concepts for linear beam theory with arbitrary geometry and loading , 1998 .

[6]  Walter D. Pilkey,et al.  BEAM STIFFNESS MATRIX BASED ON THE ELASTICITY EQUATIONS , 1997 .

[7]  N. Muskhelishvili.  Some basic problems of the mathematical theory of elasticity , 1953 .

[8]  J. B. Kosmatka,et al.  Flexure-torsion behavior of prismatic beams. I - Section properties via power series , 1993 .

[9]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[10]  I. S. Sokolnikoff.  Mathematical theory of elasticity , 1946 .

[11]  J. Jensen.  On the shear coefficient in Timoshenko's beam theory , 1983 .

[12]  Hatem Zenzri,et al.  A numerical method for the exact elastic beam theory. Applications to homogeneous and composite beams , 2004 .

[13]  J. R. Hutchinson.  Shear coefficients for Timoshenko beam theory , 2001 .

[14]  Walter D. Pilkey,et al.  On the shear deformation coefficient in beam theory , 1994 .

[15]  Morton Lowengrub,et al.  Some Basic Problems of the Mathematical Theory of Elasticity. , 1967 .

[16]  Werner Wagner,et al.  Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections , 2001 .

[17]  E. Sauer.  SCHUB UND TORSION BEI ELASTISCHEN PRISMATISCHEN BALKEN , 1980 .

[18]  Determination of centers of flexure using the boundary element method , 1993 .

[19]  Werner Wagner,et al.  Shear stresses in prismatic beams with arbitrary cross‐sections , 1999 .

[20]  E. Trefftz.  Über den Schubmittelpunkt in einem durch eine Einzellast gebogenen Balken , 1935 .

[21]  J. Goodier.  A Theorem on the Shearing Stress in Beams with Applications to Multicellular Sections , 1944 .

[22]  Walter D. Pilkey,et al.  Analysis and Design of Elastic Beams: Computational Methods , 2003 .

[23]  John B. Kosmatka,et al.  Torsion and flexure of a prismatic isotropic beam using the boundary element method , 2000 .

[24]  S. Timoshenko,et al.  Theory of Elasticity (3rd ed.) , 1970 .

[25]  W. T. Tsai,et al.  On the Determination of the Centers of Twist and of Shear for Cylindrical Shell Beams , 1972 .

[26]  A. Weinstein.  The center of shear and the center of twist , 1947 .

[27]  Evangelos J. Sapountzakis.  Solution of non-uniform torsion of bars by an integral equation method , 2000 .

[28]  R. Woerndle.  Calculation of the cross section properties and the shear stresses of , 1981 .

[29]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[30]  C. Weber.  Biegung und Schub in geraden Balken. , 1924 .