Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications

Abstract

This two-part contribution presents a beam theory (BT) with a non-uniform warping (NUW) including the effects of torsion, and shear forces and valid for any homogeneous cross-section made of isotropic elastic material. In part I, the governing equations of the NUW-BT has been established and simplified-NUW-BT versions has been deduced, wherein the number of degrees of freedom is reduced. In this part II, these theories are used to analyze, for a representative set of cross-sections (CS) (solid-CS and thin-walled open/closed-CS, bi-symmetric or not), the elastic behavior of cantilever beams subjected to torsion or shear-bending. For bi-symmetrical-CS, torsion and shear-bending are analyzed separately: analytical and numerical results are given for the distributions along the beam axis of the cross-sectional displacements and stresses, for the NUW-BT and its simplified versions. Numerical results are also given for the three-dimensional stress distributions close to the embedded section: the stress predictions of the NUW-BT are compared to those obtained by threedimensional finite elements computations. It can be drawn from all these results indications that can help to decide when the simplified theories may be applied, and hence when the warping parameters may be reduced. As specified in NUW-BT, torsion and bending are coupled for non-symmetrical-CS, even if the bending moments refer to the centroid while the torsional moment refers to the shear center. To illustrate this coupling effect, the particular example of the channel-CS presented in Kim and Kim [Kim, N.-I., Kim, M.-Y., 2005. Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects. Thin-Walled Structures 43, 701-734.] is analyzed and the results are compared.

Much ado about shear correction factors in Timoshenko beam theory

“…The arrow indicate the direction of the transverse force. Similar plots are seen in El Zenzri (2004) andEl Fatmi (2007b). A related plot is seen in Timoshenko and Goodier (1970, pp.…”

Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory

“…For the mathematical characterization (equations and boundary conditions) that governs these operators see the details (El Fatmi and Zenzri, 2002, section-3) for a 2D-characterization (on the cross-section) and (El Fatmi and Zenzri, 2004, section-3) for 3D-characterization (on a slice of the beam). For a quick description of the numerical method used to compute these operators see section-4.1 in part II (El Fatmi, 2007). …”

“…To illustrate this theory, we study in part II (El Fatmi, 2007) the elastic behavior of cantilever beams 5 subjected to torsion or shear-bending, for a representative set of cross-sections (CS) (solid-CS and thin-walled open/closed-CS, bi-symmetric or not). The comparison of the results of NUW-BT with those of simplified-NUW-BT can help to decide when simplified theories can be used, and hence when the warping parameters may be reduced.…”

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

“…During the past few years, the linear static and linear free vibration analysis of shear deformable bars undergoing twisting or general deformations have been 4 thoroughly studied [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and theoretical statements and/or numerical comparisons between beam theories ignoring or taking into account the STMDE have been presented [6-7, 9-10, 19]. However, this is not the case for vibration analysis of bars taking into account both shear effects and geometrical nonlinearities.…”