FEA Puzzler: a Tale of Two Analysts - The Solution
This FEA puzzler, published in the April issue of Benchmark, was devised by Barna Szabo of Engineering Software Research and Development, Inc.
We had a number of great responses to the challenge (business card holders are now in the post) but very few respondents spotted the singularity that Analyst B should have identified. A mesh Convergence Study (see the following extract from NAFEMS Publication "Don't Forget the Basics" by Mark Chillery /join/resources/knowledgebase/001/" could have been used to spot this problem. We enjoyed one response so much from regular NAFEMS contributor Jack Reijmers that we have shared it below!
A composite ring was made by joining a stainless steel(ss) and a titanium (Ti) ring. At room temperature (20°C) the rings are stress-free.Two analysts were asked to compute the maximum principal stress (σ1) max when the temperature of the the composite ring is increased to 120°C. They were asked to assume that the two materials are perfectly bonded and the assumptions of the linear theory of elasticity are applicable.The geometric and material parameters are shown in Table 1.
Table 1: Geometric and Material properties
Analyst A formulated this as a problem of plane stress.He used two quadrilateral elements on a 30-degreesector. The analyst verified his solution by performing a mesh convergence study and reported that the maximum principal stress occurs along the material interface and its value is 22.1 MPa.
Figure 1: Notation
Analyst B formulated this as an axisymmetric problem.He generated a uniform mesh of 800 nine-node quadrilateral elements. He reported that the maximum principal stress occurs at r = rm, z = t/2 and its value is 52.4 MPa. Relying on his judgment that the mesh was fine enough, he did not perform solution verification.Your supervisor asked you to find out why such a large discrepancy exists between the two results. Having checked the input data, you found no errors.Your supervisor would like to know what the maximum principle stress in the component is.
Analyst A solved the problem in the plane of symmetry(z = 0) and thus his model does not account for the singularity caused by the abrupt change in material properties in the points r = rm, z = ±t/2. He solved the wrong problem very accurately.Analyst B solved the correct problem, but with a very large (in fact infinitely large) error in the computed value of (σ1)max. The exact value of (σ1)max is not finite in the singular point and hence the absolute error in the reported value of (σ1)max is infinitely large.You are now faced with the unpleasant problem of having to explain to your boss that asking for the maximum stress under the assumptions of the linear theory of elasticity made no sense.
Response - Jack Reijmers
Before diving into the mechanical aspect of this problem there is also the puzzler of the background. For instance, why is the supervisor asking me? Are we dealing with a manager that is employed based on his academic level of thinking, with a complete lack of expertise in structural engineering? This is not uncommon in the modern enterprise, where management should have vision. A clear mission is more important than fundamental knowledge.Perhaps, in this case, the reason is the need for an independent party. The analysts are anonymously referred to as A and B to cover the friction involved.
The solution with plane stress elements is produced by Nathaniel Humphries (not related to) who has ample experience in structural engineering with the necessary qualifications. However, he seems distracted lately and that clearly reflects on his work. Not so strange since he discovered that his wife is having an affair with her yoga teacher. And then there is the young and ambitious Santiago Delgado who presents a solution with the application of axisymmetric elements, convinced of being on the right track. If not handled carefully this could turn into a Wild West scene. When two men go out to face each other, only one returns.A delicate problem that requires subtle formulation of the verdict to keep both analysts on board, gladly removed from the plate of the supervisor.
The solution given by Mr Humphries is given in figure 1. The titanium and steel part show a uniform stress distribution with a principal stress, ρ1 = 22.1 [MPa] in the titanium ring. This matches the value presented in the problem description. To be more precise the inner, steel ring is restrained by the titanium outer ring. This results in compression, σhoop = -22.1 [MPa].The outer, titanium ring is stretched by the expanding inner ring, giving tension, σhoop = 22.1[MPa].
The solution given by Santiago Delgado is given in figure 2. The problem description presents a principal stress of 52.4 [MPa]. The replay given in figure 2 is produced with ANSYS and by no means could the given principal stress be retrieved. However, the problem description mentions the use of a nine-node quadrilateral element and this type is not available within ANSYS.Figure 2(b) shows stress values in hoop and axial direction. Similar to the result in figure 1(b)the inner ring is in compression and the outer ring in tension in circumferential direction.
The problem has numerous angles of attack. In circumferential direction the stress levels areindifferent and therefore the Delgado-solution is valid. In fact, this also follows from the sector solution of Mr Humphries which is uniform in hoop direction.The problem definition clearly states that the rings are perfectly bonded and this essential forthe Delgado-solution. The inner ring pushes against the outer ring resulting in a major hoopstress, besides a radial stress. However, without bonding, i.e. frictionless sliding of the tworings, there is no stress in axial direction. Figure 3(a) shows the axisymmetric model where the nodes at the contact surface are only coupled in radial direction. The rings slide in axial direction without friction and the hoop stress matches the solution presented by Mr Humphries.
Probably due to an occupied mind Mr Humphries missed the condition “perfectly bonded” and therefore basic mechanical behaviour slipped from his attention.There is no load in axial direction and a homogeneous ring should be stress free in that direction. Nevertheless, the Poisson effect induces deformation in axial direction and that justifies the way of thinking of Mr Humphries. The plane strain model would be wrong and plane stress seems logical.Unfortunately, the ring is not homogeneous. Free sliding would save you from the hook, but the difference in Young’s modulus and Poison’s ratio results in a difference in axial deformation between the steel and titanium ring, see figure 4. Perfect bonding raises an interaction between the two rings and the stress free situation in axial direction vanishes. The result is a stress distribution given in figure 2(b). In short: the plane stress approach is not valid.It is sad to say but a bit more attention would have saved Mr Humphries from the pitfall. Santiago Delgado presented the correct approach, although exact stress levels could not be reproduced.