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Equivalent force distribution

In the finite element method, the equilibrium equation {f}=[K]{d} is solved for displacements {d} at the nodal locations. Hence, both the applied forces {f} and the stiffness {K} are required at these nodal positions. For example, the assignment of a constant body force (force per unit volume) requires that a transformation be performed from the volume-based loading to a set of equivalent concentrated loads that are then applied to the nodes – hence the force vector is commonly termed the equivalent nodal load vector.

Such force vectors are also termed “consistent” because the same assumptions (shape functions, integration order etc.) are used as in the generation of the stiffness matrix. That is, the stiffness and the force are consistent with each other. The only exceptions to this are the continuum solid elements and section 5.1 of the theory manual (“General Load Types”) should be consulted for further information

This section describes the way in which the finite element assumptions are used to convert a force on an element into equivalent nodal loading, specifically, the application of a uniformly distributed load to a 3-noded bar element.

The equivalent nodal force load {r} for such a uniformly distributed loading {p} is given by


Where [N] are the shape functions and S is the area over which the uniformly distributed load integration is to be performed.

In the case of a 3-noded bar element, the axial distributed load {p} is applied as follows


and in vector form as

If t is the width of the bar and dx, an incremental length along the bar, then the incremental surface area is




So that


Therefore, changing the integration limits from the local to the natural coordinate system, the equivalent nodal force is given by


Using the quadratic shape function for this bar element


This integral is accurately evaluated using a two-point Gauss integration rule in which the weight is 1 and the optimum sampling points are . So that


Or finally


Thus a constant, uniformly distributed load applied along the length of a 3-noded bar element is transformed into equivalent nodal loads which are distributed according to the ratio . If the equivalent nodal forces (px1*tL), (px2*tL) and (px3*tL) are denoted by the constant f, then, diagrammatically this distribution becomes


The same distribution is obtained for uniformly distributed edge loading on an 8-noded, 2D plane element of side length (L) and thickness (t), as follows

A constant, uniformly distributed load {p} applied to the surface of an 8-noded surface element can be similarly examined to observe the ratio of the corner and midside nodes to be , diagrammatically shown as follows

In this case, f = A*p, where (A) is the surface area of the element.

Because the element stiffness and forces are consistent with each other, this apparently “incorrect” distribution is actually entirely valid and, indeed, essential.

A number of implications arise from the use of the equivalent load vector in finite element analyses and are considered below.

Implication:  Equivalent Loading Across Multiple Elements

The foregoing discussion has been based on a higher order element. The use of lower order elements does not produce such an anomaly and the load is distributed equally between the nodes of the element.

When elements are combined in a mesh and loading applied across multiple higher order elements the same distribution takes place. It is worth noting, however, that the nodes common to the adjoining elements receive a force contribution from each of these elements. As an example, consider the diagram below in which two, higher order elements are subject to a constant, uniformly distributed load (p), as before.

In this case the centre node receives an equivalent force contribution from the two adjoining elements of (2f*1/6)). The same is true for a linear element assemblage as follows


If a summation of element reactions is performed over a single element then this additional contribution from the adjacent element(s) would need to be subtracted to be absolutely correct.

Implication:  Loading Attribute Visualisation Arrows In Reverse Directions

With a model load case active, select the attribute properties and then the loading tab. Press the button marked “settings…”, where the following two options will be seen

  •  “Show discrete loading by definition”
  •  “Show discrete loading by effect on mesh”

By default the latter is invoked and will show the equivalent nodal forces as described above, whilst the former will visualise the forces as defined in the attribute definition without any additional transformation into equivalent loads. This explains why visualising the loading attributes on a mesh, can produce a series of arrows that either alternate unexpectedly in size or reverse in direction.

Discrete loads are not the only loading types that are processed in this way for higher order elements; it applies equally to practically all loading types.

Implication:  Planar Displacement Fields

Applying “Concentrated Loads” to the edge of a higher order element to produce a uniform and planar displacement of all the nodes on that edge requires that the concentrated loads be applied in the ratios given above.

The “Global Distributed” loading also applies concentrated loads directly to the nodes in the same way that selecting the “Concentrated” load tab would, however, in the former case, the element shape functions are accounted for and the distribution handled automatically.

All other loads in LUSAS are also handled automatically.

Implication:  Reactions Are Unexpectedly Not Constant Along A Supported Boundary

Support reactions are also distributed according to the same principles. Hence the constant nodal reactions that would be naturally expected from the application of a uniformly distributed load would actually be distributed in the ratios given above.

The sum of the reactions over a single element, however, will accurately represent the total reaction acting over the length/area of the element.

Implication:  Equivalent Nodal Loading For Beam/Shell/Plate Elements

This equivalent nodal load transformation (or decomposition) must produce kinematically equivalent nodal forces from the applied element loads for elements that support moment output. For such elements, this will mean that a constant uniformly distributed load over the length of the element will produce both an equivalent shear force and a bending moment.

Kinematically equivalent loads are so named because they replace a distributed load so that the correct work is maintained. To replace the distributed load by statically equivalent forces would be incorrect and could result in errors when solving for the displacements – especially in coarse mesh definitions.

In MODELLER, kinematically equivalent loads are calculated for the relevant elements automatically.

For example, a constant value of uniformly distributed load (P) on a single beam element of length L has a kinematically equivalent nodal load defined as follows


Whereas, the statically equivalent load is defined as follows


It is for this reason that visualisation of loads in MODELLER can display additional moments at nodes on which a uniformly distributed load has been applied.

Discrete loading is an important exception to this general rule of decomposition. Discrete loading is applied to the finite element mesh and is converted into equivalent nodal loads using the shape functions of the elements. These nodal loads are then applied directly to the underlying structural mesh. Although these nodal loads correctly represent the vertical and in-plane components of the applied loading, they do not account for any kinematic decomposition. The effect of this is mesh dependent and, in the general case, is not an issue. For very coarse meshes this assumption may cause the results to be affected.

 Implication:  Joining Lower And Higher Order Elements

 It is tempting on occasion to try to join lower and higher order elements together, typically when attempting to generate a mesh transition from higher order elements to low order element. For instance, consider transitioning from 4-noded to 8-noded quadrilateral elements as follows

To demonstrate the effect of mixing elements in such a manner, consider the above situation in which a prescribed displacement of unity is applied to one end of the structure with support conditions as follows

 This loading is expected to produce a planar response throughout the structure as shown in the top-most diagram (below) where two 8-noded quadrilaterals are used. The lower of these two diagrams is the response when mixing the element types. Both diagrams show the nodal displacement magnitudes in the X-direction.


The reason for the non-planar displacement field obtained may be seen by considering the equivalent nodal force distributions required for both the 4-noded and 8-noded elements to transmit such a planar force. The following diagram shows the distributions at the element interface


For the expected results to be obtained the distribution needs to be identical. If the difference in the distribution ratios is considered we have


Although the total force across the interface will be correct, the distribution of the force will not be. That is, the difference in the ratio at nodes 1 and 4 produce a net difference in the loading of f/12, whereas nodes 2 and 5 produce a net difference of f/6 in the opposite direction. The direction of these force ratio differences correlate with the non-planar displacements that are seen, that is nodes 1 and 4 displace in the same direction as the net f/12 force and nodes 2 and 5 displace according to the direction of the f/6 net force.

If it is necessary to join elements in such a manner, it is recommended that constraint equations be used to ensure that the nodes along the element interface are constrained to displace in a planar manner.

 In summary, the joining of low and high order elements in such a manner is a dubious practice, to be avoided if at all possible and is discouraged on the basis of the different nodal force distribution (as well as stiffness) associated with the different element types which may cause inaccuracy in the results. Together with this inaccuracy the possibility of exciting an element mechanism in the 8-noded element when reduced integration is being used is greatly increased.

Finite Element Theory Contents

Local Coordinate System

Shape Functions

Numerical Integration

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