Does the location of the supports matter when using the 3-2-1 Method to apply boundary conditions to a Finite Element Model?

(a question raised at the Practical Stress and FEA course in KL, Malaysia).

The 3-2-1 Method is highly recommended, by NAFEMS and others, for linear stress analysis in 3D solid modelling. It uses the minimum constraints needed to prevent rigid body motion (large unconstrained displacements or rotation) without restricting deformation of the part in any way and produces conservative (maximum) stresses.

The 3-2-1 Method requires a balanced loading of the part, i.e. the sum of the forces and moments applied must be zero. The analyst is required to determine these, typically, using a free body diagram, so that the part is in static equilibrium. With balanced loading the reaction forces at the fixed boundary conditions (i.e. the supports) will be at or very close to zero. The 3-2-1 method is self-checking, because if large reaction forces occur then it signals that the applied external forces are not in balance and need to be corrected by an equal and opposite amount, to remove the erroneous reaction forces and moments. Note: whilst it is the ideal case that the reaction forces at the supports are zero, in practice even if the calculations of the free body forces is perfect, there will be some numerical "noise" that leads to small forces at the supports. In a static analysis even tiny forces will cause large free body translations because the part has no effective mass (within the calculation). In practice provided the forces at the supports is small in comparison to the global forces and does not produce a local stress then it is acceptable.

Only three supports are needed for the 3-2-1 method, but correct placement is essential. The first support fixes one node of the mesh in space, constraining it in the X, Y, Z directions (3 global freedoms), the second point lies on a local axis originating from the first point and constrains the node in the normal directions, i.e. the Y and Z directions, assuming the node lies on the X-axis. The third point lies in a plane passing through the first two points and prevents rotation, i.e. if the node lies in the XZ plane it will be constrained in the Y direction. A simple test that the model is correctly set up is to perform (or imagine) a thermal analysis and check that it will allow free thermal expansion without generating forces at the supports.

Figure.1. Placement of Minimum Constraints (courtesy of Tony Abbey / NAFEMS Practical Introduction to FEA)

The use of Minimum constraints with the 3-2-1 Method is one of the many useful techniques taught in NAFEMS FEA courses and was thoroughly discussed during the Practical Stress and FEA course held in Kuala Lumpur (November 2018).

The course instructor, Bon Johnson, stated that the essential requirement was the relative geometric position of the supports as described above; the absolute placing was less important (provided suitable nodes were, or could be made, available). Bob conjectured that it was even possible (although for reasons discussed below not necessarily advisable) for the boundary condition to share a loaded node. This caused some discomfort among the audience and Bob reassured them that even some seasoned analysts would feel the same. However, to prove the point, Bob provided a comparison of results for a model, which in one case had a support at a loaded node and in the other supports located away from the loads. Reflecting Bob's cycling interest, the model was based on a bicycle wrench used to adjust a threaded headset. 

Figure.2. A wrench for adjusting a bicycle threaded headset

It is reasonable to assume that this part can be modelled as a 2D plane stress problem, as the wrench is stamped from a fairly thin steel plate. For 2D problems a variation of the 3-2-1 method can be used as shown in the table below and only two supports are needed.

For Spanner-1, the node at point 3 has X, Y freedoms constrained and at point 4 the X freedom is constrained (preventing rotation in the Z axis in this case), but allowing free expansion in the XY plane. Both supports are away from the loads. In Spanner-2, the X, Y freedom are constrained by the node at point 2 and at the node at point 1 the Y freedoms is constrained, again preventing rotation in the Z. For Spanner-2, the node at point 2 is also part of the distributed loading which represents the (assumed) contact pressure on the bolt. Despite the node having both a load and a constraint it has no effect on the results, which are the same, just as Bob said………

Figure.3. Comparison of the results for different placement of Minimum Constraints (courtesy of Bob Johnson)

The results may seem counterintuitive and some mental readjustment may be needed by an analyst who is new to the 3-2-1 Method. To illustrate this, consider a simple rod loaded in uniaxial tension, being stretched by some force acting at both ends in opposite directions. It is obvious that there shouldn't be any global translation, as the forces are balanced. So, there will be a point exactly in the middle of the bar which does not move, i.e. at the point of symmetry. Now if we set up this problem using Minimum constraints and without making use of symmetry it is logical to use the point of symmetry as the location for the first support. However, using the 3-2-1 Method the supports can be placed anywhere within the bar and the deformed shape, strains and stresses will all be the same. But what will happen if the first point is located at one end of the bar, off-centre, instead of being in the centre. The bar simply translates about this point to satisfy the boundary condition at the supports without any forces being generated. There will be a net global displacement of the bar, but this will not affect the stress and strain results. Once this is appreciated it can be seen that even if the support is on a loaded face, theoretically at least it will not affect the results.

Figure.4. Effect on displaced position (illustrative) from different supports.

Point proven? Possibly, but in practice most analysts will try to avoid applying forces and boundary conditions on a shared node. As well as the possibility that this could cause errors, there are several good practical reasons to space the supports well apart and away from regions of interest (in addition to being in the right relative geometric location). A greater spacing increases the moment arm and reduces the reaction forces at the supports needed to resist turning moments, due to small out of balance forces. By placing the supports in a low stress region, erroneous reaction forces at the supports will be highlighted by a local hot spot in the stress and easily spotted in the contour plot. If the supports are within areas of high stress, this local stress at the support will be masked (however checking the reaction forces at the support nodes should be done anyway and will pick this up).

A limitation of the 3-2-1 method is that it is not applicable to non-linear geometry cases. Even for linear problems it may be challenging to calculate balanced forces using free body diagrams, especially for complex load cases. It can also be a problem to determine natural distributed loads at the loaded faces (point forces should be not used). Typically a pressure distribution of various analytical forms will be used, but it is necessary to check whether the results are sensitive to this assumption. This is more likely to be an issue if the maximum stress occurs near to the loaded surface.

Using a non-linear model allows more real-world physics to be included. In the case of the wrench discussed an alternative approach would have been to include the bolt in the analysis and use contact modelling to calculate the contact forces between it and the wrench. Including non-linear geometry and plasticity, would allow the engineer to identify failure cases, for instance whether the bolt could start to rotate in the wrench due to deformation at high loads.

Even when minimum constraints is a suitable approach, many users of modern FEA tools may feel that it is just simpler and faster to set up a model using contact, as the additional computation cost is unlikely to be of any concern. However, there is still a place for minimum constraints, by providing natural loads the results are often very clean without spurious areas of high stress that need to be explained away. Even for complex situations it can be a useful stage in the validation process building up to a full scale analysis.

Figure.5. Bob testing his riding skills and the bike - having confidence in Engineering Simulation is increasingly important.

References:

• Terminology: https://www.nafems.org/publications/glossary/
• A review of the 3-2-1 Method: https://www.eng-tips.com/viewthread.cfm?qid=239491
• Article in Digital Engineering (Tony Abbey): https://www.digitalengineering247.com/article/free-floating-fea-models
• Further training (Tony Abbey LinkedIN course): https://www.garudax.id/learning/solidworks-advanced-simulation/description-of-the-3-2-1-method