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

Section

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 a_{1} and a_{2} are constants and u is the displacement at some
position, x, along the element: u = a_{1} + a_{2}x

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 = a_{1} + a_{2}x + a_{3}x^{2}

Which when differentiated gives a linear variation in strain: a_{2} + a_{3}x

**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)x_{1} + 0.5(1 + r)x_{2}

but as x_{1} = 0 and x_{2} = 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 |
||

c_{1} |
0.555555556 |
r_{1} |
-0.774596669 |

c_{2} |
0.888888889 |
r_{2} |
0 |

c_{3} |
0.555555556 |
r_{3} |
0.774596669 |

Calculating the extension, delta:

Further improvement is obtained with four point Gauss integration as shown below:

Weighting factor |
r value |
||

c_{1} |
0.347854845 |
r_{1} |
-0.861136312 |

c_{2} |
0.652145155 |
r_{2} |
-0.339981044 |

c_{3} |
0.652145155 |
r_{3} |
0.339981044 |

c_{4} |
0.347854845 |
r_{1} |
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.