### Minimum Constraints in FEA – 3. Zero Resultant Force Problems by Kotur Raghavan

General

Objects like floating bodies and vehicles in flight have two distinct features. In the first place, they are subject to self-equilibrated system of forces. Secondly, they will have no identifiable displacement constraints. Problems of this class can be solved by applying minimum constraints. This aspect will be demonstrated by way of a few case studies.

Perforated Disk under the action of uniform Circumferential Pressure

The object shown in Fig. 1 is the first case study problem. Plane stress analysis is carried out.

Fig. 1

Two independent analyses are carried out for two different sets of displacement constraints. The constraint sets C1 and C2 are shown in Fig. 2. In each case the rules are adhered to.

Fig. 2

The nodes chosen for constraints are arbitrary. We can expect the deformation patterns to be different for the two different constraint sets. This is confirmed by the contours of resultant deformations shown in Fig. 3.

Fig. 3

The stresses, however, are not governed by absolute deformations. Strains and stresses are functions of relative deformations. For both the constraint sets, the stress contours happen to be identical. This can be observed from the contours of von Mises stress given in Fig. 4. This is one check to ensure that the constraint are applied correctly.

Fig. 4

There is another check to verify that the constraints are correctly applied. This is by studying the reaction forces at the constrained degrees of freedom. As the applied forces are in self-equilibrium, the resultant force is zero and hence the total reaction forces will be zero. This is irrespective of whether the constraints are properly applied or not. In addition, however, the requirement is that the reaction forces at each of the constrained DOF need to be zero.

The computed reaction forces for the constraint set C2 are as give the above box. The numbers, which are negligibly small, confirm that the constraints are appropriate.

It would be interesting to look into the displacement patterns for different constraint sets. In Fig4A below the variations of X- displacement along the horizontal diameter are plotted.

Fig. 4A

The two plots are running parallel to one another with a “rigid body” shift. The slope, or in other words the derivatives, will be same. Consequently, the strains and the stresses remain unaltered.

The same object is analysed for hydrostatic loading condition. Pressure (unit value) is applied on the inner surfaces of the cut-outs. The applied pressures are as indicated Fig. 5.

Fig. 5

According to theory, the body will experience uniform compressive stress along all directions. This is exactly the result obtained through analysis. This is observable from Fig. 6 which contains the stress fields of normal stress in X- direction and von Mises stress.

Fig. 6

These results are obtainable only by imposing the minimum constraints.

Transverse Fillet Welded Components as a Free Body

In an earlier article, design analysis of plates joined by transverse fillet welds was carried out.  See https://www.fembestpractices.com/2020/11/design-procedure-review-02-transverse.html . In the analysis model, only one half of the assembly was analysed and symmetry condition was used. Presently the full model will be analysed as a free body and imposing minimum constraints. The model for analysis is shown in Fig. 7.

Fig. 7

As per the procedure followed in the safe value of F was found to be 1852 newtons. Analysis is carried out for the same value of load as in the earlier analysis. Displacement constraints are imposed as indicated in the figure.

Fig. 8

The computed stresses in the weldments are shown on the left hand side in Fig. 8. They are exactly same as those obtained from the symmetry model (RHS).

Other Practical applications

We come across many practical problems wherein we can exploit the principle of minimum constraints to tackle constraint uncertainties. Floating bodies belong to this class.

Fig. 9

The simulation approach for this class of problems is defined in Fig. 9. The problem is that of a floating rectangular vessel. Hydrostatic pressures on the side walls will be in equilibrium. The buoyant force on the bottom plate will be in equilibrium with the gravity force due to self-weight and the additional point load at the centre. The problem is analysed by imposing the constraints as shown.

Fig. 10

The results of analysis are shown in Fig. 10.

In the figure below is shown a column which is under compression caused by tensioned tie-rods.

Fig.11

The tensile force in the tie rods will be in equilibrium with the compressive force in the central column. This is another class of problem which can be solved by imposing minimum constraints.

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1. This is nice!

2. This is regarding your Figs. 3 and 4. You have plotted USUM. It would be interesting to see contours of u and V. Two different sets of functions [ u1(x,y),v1(x,y)] and [u2(x,y),v2(x,y)] yield identical derivatives (du1(x,y)/dx is same as du2(x,y)/dx and sp on). It would be interesting to see these two plots on some representative paths and compare to see what binds them to yield identical derivatives

3. I will share the paths.

4. The displacement patterns for two different constraints sets happen to be parallel to one another, Therefore , the slopes will be exactly same and hence the strains and stresses will same. I will include a figure in the text and add description. Thanks.