## Errors Arising in FEA Example: a Tapered Bar in Tension

Section A calculates the extension of the tapered bar analytically.

The results in section B were worked out to simulate the effect of using different numbers of constant strain elements for FEA of the tapered bar.

In C one element is mapped and numerical integration is carried out using the Gauss method for 2, 3 and 4 points.

A   Increasing the number of simple elements increases accuracy. Alternatively accuracy can be improved by using a smaller number of more complex elements - where the displacement and shape functions are represented by a higher order polynomial.

Inspection suggests that the stress and strain along the bar will not be constant, but will vary in a linear manner, so for a better result using fewer elements, such variation must be available in the elements.

Assuming a1 and a2 are constants and u is the displacement at some position, x, along the element: u = a1 + a2x

Strain is the differential of the displacement: The above linear variation in displacement gives constant strain and stress.

To have a linear variation in strain and stress requires a quadratic displacement polynomial: u = a1 + a2x + a3x2
Which when differentiated gives a linear variation in strain: a2 + a3x

C

For a 1 d element with a quadratic polynomial modeling the displacement, analytical integration of the equation as carried out at the top of this page is not needed. By mapping the above element which runs from x = 0 to x = 1 to the standard format of running from r = -1 to r = 1, Gauss integration can easily be carried out by determining the value of the equation at the 2 Gauss points and summing the values.
The Gauss points are located at r = -0.577350269 and r = +0.577350269 and weightings of unity are used. Mapping: x = 0.5(1 - r)x1 + 0.5(1 + r)x2

but as x1 = 0 and x2 = 1, then x = 0.5(1 + r)

Differentiating gives: dx = dr/2 Numerically evaluating the integral by using the value of r at the two Gauss points: - and + 0.577350269: This gives an extension of 0.09782608mm

The result can be further improved by using the Gauss method with 3 integration points:

 Weighting factor r value c1 0.555555556 r1 -0.774596669 c2 0.888888889 r2 0 c3 0.555555556 r3 0.774596669

Calculating the extension, delta: Further improvement is obtained with four point Gauss integration as shown below:

 Weighting factor r value c1 0.347854845 r1 -0.861136312 c2 0.652145155 r2 -0.339981044 c3 0.652145155 r3 0.339981044 c4 0.347854845 r1 0.861136312 The results are summarised below:

 Theoretical extension for actual geometry 0.10059mm extension 1 element 60mm wide 0.08333mm extension 2 elements, each 0.5m long, 80mm and 40mm wide 0.0937mm 4 elements, each 0.25m long, 90, 70, 50, 30mm wide 0.0984mm 8 elements, each 0.125m long; 95, 85, 75, 65, 55, 45, 35, 25mm wide 0.09999mm 2 point Gauss integration 0.0982608mm 3 point Gauss integration 0.1001684mm 4 point Gauss integration 0.1005269mm

By specifying nodes between the ends of beam and bar elements (and between the corners of quadrilateral and hexahedron shaped elements) curved components and components with curved edges and surfaces can be modeled.
By carrying out a suitable mapping transformation, elements with straight and curved edges can be transformed to the standard format for lines of extending in a straight line from r = -1 to +1, for quadrilaterals extending in a square from r = -1 to +1 and s = -1 to +1 and for hexahedrons to a cube extending from r = -1 to +1, s = -1 to +1 and t = -1 to +1.
Numerical integration with comparatively high accuracy is obtained when using the Gauss approach, see above, so these procedures are commonly used in fea software. Using this approach also makes it comparativley easy to develop and add new elements to existing software. The main disadvantage of this approach involves the mapping which uses the Jacobian matrix where difficulties can arise if elements are initially very distorted. Modern software packages check for these potential difficulties and issue appropriate warnings.

David Grieve, 1st October 2012.