_{1}

In structural analysis, it is often necessary to determine the geometrical properties of cross section. The location of the shear center is greater importance for an arbitrary cross section. In this study, the problems of coupled shearing and torsional were analyzed by using the finite element method. Namely, the simultaneous equations with respect to the warping, shear deflection, angle of torsion and Lagrange’s multipliers are derived by finite element approximation. Solving them numerically, the matrix of the shearing rigidity and torsional rigidity is obtained. This matrix indicates the coupled shearing and torsional deflection. The shear center can be obtained determining the coordinate axes so as to eliminate the non-diagonal terms. Several numerical examples are performed and show that the present method gives excellent results for an arbitrary cross section.

Beam theory is often used in the analysis of many structures in the initial design stage, including machinery, ships, and vehicles. Because beams are three-dimensional (3D) elastic bodies, their behavior must be calculated on the basis of 3D elasticity theory if it is to be calculated precisely. However, there are several difficulties involved in calculating an exact solution. Thus, current beam theory is systematized under the assumption that beams are long, thin structures, with cross-sectional dimensions that are very small with respect to their lengths [

One of these constants is the shear center, defined as the point at which a beam’s flexural deformation can be separated from its torsional deformation. The shear center is significant for the mechanical behavior of beams, and so it must be determined correctly in advance of analysis. Various definitions of the shear center discussed in Fung [

1) Setting the shear center as the point of action of the resultant of the shearing stresses in the section;

2) Setting the shear center as the point of the torsion-free bending on the basis of the strain energy considerations [

3) Setting the shear center as the point where, in the flexural-torsional problem, the axial force of the torsion arise [

However, it is difficult to accurately determine the location of the shear center in beams of arbitrary cross- sectional shapes; the cross-sectional shapes for which it can be determined are quite limited. Thus, a practical calculation method is sought that can determine the shear center and other cross-sectional constants for beams of arbitrary cross-sectional shape. Analysis using the finite element method (FEM) is practical and useful, and extremely beneficial in the initial design of beam structures based on beam theory. IN [

Therefore, starting with the assumption of the cross-section invariant, we show here a practical, FEM-based calculation method for determining the shear center of a beam of arbitrary cross-sectional shape. The first, we discuss the shear-torsion coupling problem for beams of arbitrary cross-sectional shape based on Saint-Venant’s theorem [

In a straight beam of uniform cross section of the type shown in

where

Next, an x-axis displacement of the beam should accompany the cross-sectional displacement given by Equation (1). This x-axis displacement is approximated over each element e by

where u is the vector of nodal values u_{i} of u for the element and _{i}.

In the following equation, the shear strains generated in the crosssection of the beam

By substituting Equations (1)-(2) into Equation (3), the shear strains

where a prime denotes derivation with respect to x, that is

The strain energy U of per unit length along the x-axis is found by evaluating the following integral:

The potential energy L for the external forces meanwhile is given as follows

where Q_{y} and Q_{z} are the shear forces in they and z directions, respectively, and M_{x} is the torsional moment.

Let us now consider how to minimize the total potential energy

The axial stress can be expressed as follows when the beam is subjected to axial forces P and moments (M_{y}, M_{z})

The warping u by shear and torsion is independent of the u due to axial force and bending

Substitute of Equation (7) into Equation (8) and the integrals of Equation (8), in terms of the warping u, become

At this time we make use of the method of the Lagrange multiplier. That is, we multiply Equation (9) by an undetermined constant

The stationary condition for _{1} in the above equation becomes the following:

Thus,

in which

Writing an inverse matrix of the left-hand-side matrix of Equation (12) as

Thus, according to Equation (12)

From Equation (15), the following relationship can be derived easily

In addition, the warping u in the x-axis generated by shear force and torsional moment can be obtained via Equation (15)

Converting the above displacement of and force acting on the centroid to an arbitrary point (y_{s}, z_{s}) in the cross- sectional plane (e.g.,

Now replacing the relationship between shear force and shear deflection on the centroid in Equation (16) Using Equation (18), we arrive at the result:

where

The shear center is obtained by equating to zero the elements (1, 3) and (2, 3) of matrix D_{s}, i.e.

Where D_{ij} is the (i, j)^{th} element of

The procedures of the previous sections have been applied to two example problems. Calculations were performed using triangular linear elements, which are widely utilized in typical finite element analysis.

Example 1. Initially selecting the semicircular cross section of radius R as shown in

The shear center (y_{s}, z_{s}) is thus derived as

The function

where

These results differ from the results of our FEM-based analysis by less than 1%, suggesting our solution has sufficient accuracy in practice.

Example 2. Determine the shear center of a circular cross section with a circular notch shown in _{s} and centroid y_{c}, for various ratios a/r of the radius r of the circular cross section versus the radius a of the circular notch. The figure also shows the results of the analsysis of Strongeand Zhang [_{x} has acted. Contour has become finer in the notch tip. This tendency is remarkable enough to a/r is smaller.

This paper discussed a practical calculation method to address shear-torsion coupling. Based on the FEM, we proposed a method to analyze the shear-torsion coupling problem when analyzing beams with cross sections of arbitrary shape as a stationary value problem with subsidiary condition. Specifically, we constructed simultaneous equations for strain deflection, torsion angle, warping, and undetermined Lagrange multipliers,

Timoshenko | Kawai et al. [ | Exact Equation (25) | Present | |
---|---|---|---|---|

Model I | Model II | |||

0.511R | 0.509R | 0.509R | 0.508R | 0.509R |

using finite element approximations. We then solved this to derive a matrix of strain rigidity versus torsional rigidity. This matrix represents the coupling of shear deflection and torsion: we successfully determined the shear center by defining a coordinate axis such that non-diagonal terms in it disappeared. We analyzed a simple example using a calculation program created based on this analysis, and confirmed the usefulness of this technique.

The author thanks the editor and the referee for their comments.

Hiroaki Katori, (2016) Determination of Shear Center of Arbitrary Cross-Section. World Journal of Mechanics,06,249-256. doi: 10.4236/wjm.2016.68020