Shear Correction Factors and Shear Center by VABS

Introduction

VABS can calculate shear correction factors for simple isotropic sections. For general sections, VABS can provide an accurate 6 X 6 1-D stiffness matrix including transverse shear stiffness for Timoshenko beam theory. The originally very difficult task of locating the shear center, which is very important for design of thin-walled open section beams, becomes trivial based on the 6 X 6 stiffness matrix. We have tried the following examples:
  1. a symmetric channel section, compared with the thin-walled theory
  2. shear center for a triangular section
  3. shear correct factor for a rectangular section
  4. shear correct factors for an arbitrary section

Example 1

For a channel section (see Fig. 1) made with isotropic material, the shear location calculated from the model of VABS normalized by the thin-walled solution is plotted in Fig. 2 as the ratio b/t varies. The mesh is different for different values of this ratio. For b/t=10, the channel section is meshed by 472 8-noded quadrilateral elements for a total of 4983 degrees of freedom. It is obvious that when the ratio is very large, the shear center location calculated by the VABS is convergent to that of the thin-walled theory. When the section is not thin enough, the thin-walled assumption will introduce significant errors.


Fig. 1 Sketch of a channel section


Fig. 2 Shear center location of channel section

Example 2

There is no known exact solution for the shear center of a general solid triangular cross section (Fig. 3). The shear center location versus the aspect ratio is plotted in Fig. 4. When b/a is very small, the triangular section acts as thin rectangular section and the shear center approaches the mid-point. For isotropic equilateral triangular section, the shear center is the centroid. When b/a is very large, the shear center moves toward the vertical edge.


Fig. 3 Sketch of triangular section


Fig. 4 Shear center location of triangular section

Example 3

It has been proven that VABS can find the same shear correction factors as those based on 3-D elasticity theory if such solutions are available (rectangular, elliptical, and equilateral trangular sections have exact solutions). This example study the shear correction factor for a rectangular cross section with width 1 inch and height 2 inches. The exact shear correction factor along the width from 3-D elasticity is 0.7884442. Fig. 5 is a log-log scale plot showing the convergence of VABS results to the exact value. When there are more than 2 elements along the width, the error is less than 1%.


Fig. 5 Convergence of VABS result to exact solution

Example 4

To demonstrate that VABS can calculate the shear correction factors correctly for arbitrary sections, an irregular section as sketched and meshed in Fig. 6 is studied. VABS results are compared with those of an analytical work (Gruttmann and Wagner, 2001) and ANSYS beam capability (beam 188/189) in Table 1. VABS results agree with the analytical results (Gruttmann and Wagner, 2001} within 0.2%, while ANSYS results are off by as much as 1.3%. Note that the results from ANSYS are independent of Poisson's ratio and thus can be only considered as approximations to the exact solution when Poisson's ration is zero. It should be noted that the work of Gruttmann and Wagner is devoted to calculate the shear correction factors only for arbitrary isotropic sections, which is much less versatile than VABS because this capability is just a small subset of the VABS functionalities.


Fig. 6 Geometry and mesh of an arbitrary cross section

Table 1 Shear correction factors for an arbitrary section
Poisson's ratio Resources00.250.5
c2/aVABS0.74040.73670.7306
GW0.73950.73550.7294
ANSYS0.74020.74020.7402
c3/aVABS0.67800.67640.6736
GW0.67670.67530.6727
ANSYS0.67780.67780.6778