When to Perform Non Linear Geometric Analysis
The following study looks at the different results obtained from a cantilever beam
undergoing large displacements deformation.
The rule of thumb is if the largest displacement in your model is greater than
10% of the largest dimension of your model than you should run a geomtric
non-linear analysis. Another way of looking at it is if your structure violates
small angle theory then you should run a non-linear geometry analysis. Small angle
theory says that that for small angles the vertical displacment equals the arclength or:
Arclength = Radius * Theta ( in radians ) =~ Sin ( theta )
For theta < 10 degrees the above holds true as shown in the calculator below
for a unit circle ( radius = 1 ) :
Enter the angle in degrees:
When the structure deforms more than 10%, the stiffness
matrix needs to be updated with the deformed geometry. Linear analysis is always based
on the undeformed geometry and deformations just scale linearly with load . Therefore,
in a non-linear geometry run, you will get much more realistic deformations that match
how the structures stiffness is changing as it is being deformed.
The simplest example to illustrate this is a cantilever beam loaded at the tip. As the
load increases the beams bends in an arc, causing the tip to actually move inwards . This
inward movement of the tip will not be picked up by a linear analysis since the 'arcing' of
the beam is dependent on the updated deformed shape .
Additionaly, as the beam arcs inwards the bending moment arm of the applied tip load decreases. As
a result the vertical displacements are less than the linear analysis due to the lower bending moment
at the higher loads.
| Axial Displacement Comparison |
| Load | Non-Linear |
Linear |
| 400 | -0.222 | 0.576 |
| 800 | -0.263 | 1.152 |
| 1200 | -1.4 | 1.728 |
| 1600 | -3.06 | 2.304 |
| 2000 | -5.11 | 2.88 |
| 2400 | -7.4 | 3.456 |
| 2800 | -9.83 | 4.032 |
| 3200 | -12.3 | 4.608 |
| 3600 | -14.76 | 5.184 |
| 4000 | -17.2 | 5.76 |
The axial results show how the linear analysis is unable to accurately predict the
arcing of the beam as it bends inwards.
| Vertical Displacement Comparison |
| Load | Non Linear |
Linear | % Error |
| 400 | -7.74 | -7.68 | < 1 % |
| 800 | -15.4 | -15.4 | < 1 % |
| 1200 | -22.7 | -23.0 | 1 % |
| 1600 | -29.6 | -30.7 | 4 % |
| 2000 | -35.8 | -38.4 | 7 % |
| 2400 | -41.3 | -46.1 | 12 % |
| 2800 | -46.2 | -53.8 | 16 % |
| 3200 | -50.5 | -61.4 | 22 % |
| 3600 | -54.3 | -69.1 | 27 % |
| 4000 | -57.6 | -76.8 | 33 % |
The vertical displacement comparison shows how the non-linear displacements
are lower than the linear analysis since the bending moment arm is decreasing as the
beam arcs inwards.
| Maximium Von Mises Stress Comparison |
| Load | Linear |
Non-linear | % Error |
| 400 | 21.8 | 22.1 | 1 % |
| 800 | 43.6 | 44.2 | 4 % |
| 1200 | 62.1 | 66.3 | 7 % |
| 1600 | 80.1 | 88.4 | 10 % |
| 2000 | 96.4 | 111 | 15 % |
| 2400 | 112 | 133 | 18 % |
| 2800 | 125 | 155 | 24 % |
| 3200 | 137 | 177 | 29 % |
| 3600 | 147 | 199 | 35 % |
| 4000 | 157 | 221 | 41 % |
The stress analysis comparison shows how significant differences in the stresses
exist as soon as the small angle theory is violated
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