A FINITE ELEMENT CODE FOR TORSIONAL ANALYSES USING PRANDTL STRESS FUNCTIONS

Paranjothy Sai Sudharsanan

VERIFICATION AND VALIDATION

UNIT CIRCLE:

For a unit circle with G=0.1 , analytical solution gives the value of torsional rigidity multiplier as 1.5707.Analytical solution for Prandtl function (Phi) is of the form -0.05(x2+y2-1). The value of Phi has been calculated at the centre to be 0.05 and referenced for solution convergence. The maximum difference between the Phi through FEM and Analytical solution were incorporated as a part of the code for the entire domain. K and Maximum of Phi-Difference values for different densities of meshes and difference between FEM Solution and Analytical solution are presented in Table1.

Number of Elements K-Analytical K-FEM Difference

Max phi-Difference

71 Elements 1.5707 1.5104 0.0603 0.0006 130 Elements 1.5707 1.5392 0.0315 0.0003 204 Elements 1.5707 1.5501 0.0206 0.0001

Table-1

Fig.1 Fig.2

The Shear stress in xz direction is analytically computed to be -0.1y and yz direction is 0.1x. Substituting G as 1, shear stress and strains become equal in the respective directions. The analytical solutions are of the form xz=-0.1y and yz=0.1x using the following relations

Phi and shear stress plots for mesh density of 204 elements have been shown in figure 3 through figure 5. For y=1 and x=-1, xz and yz are computed to be -.0941 and -.092 respectively. The slight discrepancy in values are due to the computation of stress at quadrature points and then extrapolating them to nodal values.

0

0.02

0.04

0.06

0.08

1 2 3

Difference

0

0.0002

0.0004

0.0006

0.0008

1 2 3

Difference

Fig.3 Phi Fig.4 xz Fig.5 yz

ELLIPSE:

The value of total torque for the elliptical cross section can be computed as 0.031426 and the value of K from torque as 0.31426 assuming G=0.1. The following relations were used

= +

+

1

a=1 and b=0.5 and integrated over the domain.

FEM solutions predict a value of 0.3120 for a mesh density of 412 elements, which seems fairly accurate. Plots for Phi and shear stresses have been shown in figure 6 through figure 8.

Fig.6 Phi Fig.7 xz Fig.8 yz

EQUILATERAL TRIANGLE:

Using the same relations as above for torque and K, but by replacing the Phi by Prandtl stress function for an equilateral triangle, the torque was computed to be 0.002165.K was obtained from torque as 0.02165.

=3

23 3

3 +

3 +

3 +

FEM solutions predict a value of 0.0213 for a mesh density of 99 elements. The plots have been shown in figure 9 through 11.

Fig.9 Phi Fig.10 xz Fig.11 yz

SQUARE:

The torsional rigidity factor for the square rod was calculated using the series for K. The value of the series did not show much variation when the number of terms were increased, it remained the same until the fourth decimal. Comparison of rigidity multipliers for three different mesh densities have been shown in table-2. Plots of Prandtl stress functions and shear stresses for the square section have been shown from figure 12 through figure 14.

Number of Elements No. of K terms

K-Analytical K-FEM Difference

6 Elements 3 0.14059 0.1348 0.0058 8 Elements 4 0.14058 0.1373 0.0033 10 Elements 5 0.14057 0.1385 0.0021

Table-1

Fig.12 Phi Fig.13 xz Fig.14 yz

Fig.15 Convergence of FEM Solution to Analytical Solution

DESCRIPTION OF THE CODE:

The code takes input of the nodal positions and connectivity from the ABAQUS file through readinp and readinp2 functions respectively. The elemental and system degree of freedom are computed and local and global stiffness and force matrices are initiated. The nodes where essential boundary conditions are applied are manually fed in bound_dof matrix and their respective values are set to zero through the bound_val matrix. Stiffness and Force matrices are computed in a series of for loops, where the first for loop stands for the total number of elements, the second for total number of nodes in an element, the next two are for gauss integration. Aforementioned computation is in accordance with the matrix system of equations derived earlier through the weak form.

The constraints function sets the Phi as zero on the boundary nodes and particular stiffness term related to that node is set as one. Phi, the Prandtl stress function obtained as a solution from solving global system of equations Phi = Kg\F. It is then used in the postprocessing part of the code and element stresses and torque are calculated. The torsional rigidity multiplier kr is in turn computed from the total torque.

A similar procedure is carried out with a series of loops effecting summation of terms per element and global assembly which is controlled by the index file. The stresses are computed at the quadrature points since they are discontinuous at the nodes and signs are set in accordance with the relation between shear stresses and Prandtl function. A global stress matrix,stress_gausspoint is then assembled for their corresponding global quadrature points. A uniform grid txi and txy are then formed and meshed. The stresses are then plotted to uniform mesh to give an approximate estimation of the nodal stress values through griddata command.

Salient features of the program is that it could be converted to a two degree of freedom system a few modifications since the shape functions and their derivatives are arranged in a matrix through indices. The program is also enabled to compute the time it takes to compute the solutions and write the output plots to a .pdf file. To solve a elliptic cross section with 412 twelve elements, the program takes 15 seconds on an average.

REFERENCES:

The Finite Element Method using MATLAB, Young W. Kwon and Hyochoong Bang