Stress linearization: Pm+Pb instead of Pm+b

[jura.tomorad]

When analysing stresses on pressure vessels according to code (EN 13445-3 Annex C or ASME VIII Div.2 Part 5), it is necessary to calculate equivalent membrane + bending stresses thorught the thickness of the material. According to code, it should be done in a below described manner:

1. Calculation of stress tensor regarding to local coordinate system T,N,H on first point, last point point  and all points in between (that means that the function σ_ij(T) should be acquired)

Nastran performs this step, a little bit incorrectly though (as confirmed in several posts) for parabolic elements,  and it can be seen in stress linearization output:

2. Decomposition of calculated stress functions 

σ_ij_m(T) (membrane) and σ_ij_b(T) (bending) should be calculated now using the already calculated σ_ij(T) by following the formulae from the standard. Results should look like shown below (3 matrices):

Is it possible to extract σ_ij_m(T) (membrane) and σ_ij_b(T) (bending) separately from NASTRAN?

3. Sum of membrane and bending stresses

There is always a larger and lesser sum. It is either σ_ij_m + σ_ij_b (first point) or σ_ij_m + σ_ij_b (last point). For this case, let's assume that the critical point is the first point. The larger sum is crucial for stress evaluation:

4. Equivalent sum of membrane and bending stresses

In the end, equivalent stress of the matrix Σ_ij_m+b  according to either Tresca or Von Mises theory should be calculated. Let's call it  P_m+b.

P_m+b is the stress that should be compared with the allowable stress from the standard. However, that is not what stress linearization output gives:

Stress linearization output in NASTRAN calculates equivalent strees from σ_ij_m(T) and σ_ij_b(T) separately and displayes them as Pm and Pb. But, then just calculates the sum of Pm and Pb.

Sum Pm+Pb is always larger than P_m+b because it doesn't allow for opposite signs (+/-) that are present in matrices σ_ij_m(T) and σ_ij_b(T) to have any effect.  

It can be argued that the NASTRAN solution is "safer", but it can be much to safe and give a wrong impression.

Is it possible to see P_m+b instead of Pm+Pb from NASTRAN output?

Thank you in advance for taking my question into account.

Kind regards,

Jura Tomorad

[John_Holtz · Accepted solution]

Hi @jura.tomorad 

According to the documentation, the output is Pm+b but the interface labels it (incorrectly) as Pm+Pb.

• The procedure calculates the six (6) stress tensors due to membrane+bending, m+b (equations 19 through 24)
• The principal stresses are calculated from those, (p1m+b, p2m+b, p3m+b).
• The von Mises or Max Shear stress is calculated. Since they are calculated from m+b, it would be more accurate to label the result as Pm+b.

John

[jura.tomorad]

Thank you for pointing this out. This confirms the correctness of Pm+Pb (or as it should be labeled P_m+b). However, I have two more questions:

1. Documentation states that membrane stress is calculated for all 6 stress components (σ_N, σ_T, σ_H, τ_NT, τ_TH, τ_HN) as it is expected. But it also states that bending stress is calculated for only 3 stress components σ_N, σ_H, τ_HN. Why isn't it calculated for σ_T, τ_NT and τ_TH as well?

2. In the final note, it states the following:

Pb is not important for stress evaluation according to code, but I think this is not correct. Pb is the equivalent stress of bending stress matrix. Let's consider this example (stress values are not exported from software, just randomly generated. Equivalent stresses are calculated according to Von Mises theory:

From this example it is visible when Pb is calculated as a difference of P_m+b (calculated in NASTRAN, but mislabeled) and P_m (calculated in NASTRAN) it doesn't give corresponding Pb. Also, the difference between P_m+b and P_m+P_b is visible.