The DLF (dynamic load factor) can be determined according to Annex II of ASME B31.1 and used in static analysis. An excessive but frequently used simplification is to assume a maximum DLF value of 2.0. The DLF obtained in one way or another is a multiplier of the PSV recoil force, obtained from the manufacturer or calculated independently.   

The spectral response method is used to evaluate the system's response to vibrations at different frequencies. The responses at a given frequency are combined into the overall system response using the SRSS method.  

The response history method allows the load to be scheduled at any time and place with the use of damping.

Below is an interesting video generated by Cesar II showing how the entire system behaves.  

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When is time history used? We use this method to simulate short-term, rapid events. Below is an example of analyzing a bank of 4x6-inch natural gas safety valves at pressures of 6.4 MPa/1.15 MPa. Determining the force profile:

• T=0.00 s: Force = 0 Valve closed

• T=0.01 s: Force = 35 kN

• T=0.50 s: Force = 35 kN

• T=0.51 s: Force=0 Valve closing

The AutoPipe model shows the following static analysis results. Ideally, 100% of the B31.3 code stress

A force of 35 kN is applied to the centers of the arches. This force is absorbed by the triunion supports with the LS function.

To demonstrate how the presence or absence of the first support downstream of the PSV valve affects the dynamic results, it was removed from the first set from the top. The absence of the support causes the displacement to dampen the oscillations.

Also, very large force oscillations, which significantly exceed the recoil force of 35 kN, can now be calculated as the actual DLF factor of 61 / 35 = 1.85. This shows that frequently assuming a maximum DLF value of 2.0 can sometimes be justified. In this case, it can be justified by the support failure scenario.

It's interesting to see how the oscillations develop further along the pipeline. For example, at the first elbow of the collector, the force is significantly lower, only 18 kN.

For comparison, in a situation where the support is working properly, the pipeline and the support are so stiff that no oscillation occurs.

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In thick plates, PSO occurs, during which the core of the element is "trapped" by the enormous mass of surrounding steel. The material inside wants to contract but physically cannot. This results in a triaxial stress state – the third axis in the thickness direction. In this state, shear stresses are blocked, which are responsible for "slippage" in the crystal lattice, i.e., plasticity. Since the steel cannot flow plastically, stresses increase until they exceed the tensile strength of the crystal lattice itself. This results in brittle fracture at far the lowest permissible temperature. The steel breaks suddenly and with a tremendous bang, and with minimal energy absorption, just like glass. Therefore, thicker plates force designers to choose steels with a higher guaranteed fracture energy. This rule only applies to structures operating continuously at low temperatures (bridges), so it does not apply to galvanizing tanks operating at the temperature where creep begins.

How do you assess what value of work of fracture is sufficient? The Charpy impact test is a test dynamic on a notched sample. From it, the fracture work is determined, which is included in the standards. In turn, the most important material feature from the point of view of fracture mechanics, i.e. the critical stress intensity factor at plane strain PSO (K_IC), is determined under the conditions statycznych using a fatigue crack. There is no purely analytical formula that connects these two values. However, since K studies_IC are very expensive and time-consuming, empirical relationships have been developed that allow estimating K_IC Based on the cheaper and faster Charpy test. Thanks to these formulas, having studied the work of fracture, the designer can estimate the K_IC and calculate the critical defect size (e.g., a microcrack) that will lead to the destruction of the structure or apparatus. In other words, fracture mechanics will not answer the question of whether a low fracture energy value at room temperature is the primary cause of failure. However, it could answer whether, for a given fracture energy, a given initial microcrack can develop into a full crack through the thickness of the tank sheet. However, we will not have such data.

Why doesn't the work of fracture alone determine the breaking stress? The yield strength tells us at what stress a material will crack. idealny (without defects) will begin to permanently deform. The work of fracture, on the other hand, tells us how much energy the material can absorb once it has has a defect (notch) and we hit it dynamically. The breaking stress is proportional to K_IC and to the size of the notch according to the generally known formula:

The relationship between fracture toughness and yield strength is usually inversely proportional. There used to be a relationship for structural steels that the harder and more durable the steel, the more brittle it is. Today, with sophisticated heat treatment methods and alloying additives, this relationship is no longer so obvious. There is no single, magic formula. You have to follow the path: KV – K_IC – determining the defect size – calculating the breaking stress. In the power industry, there is a strictly observed rule regarding the commissioning and shutdown of pipelines operating at critical parameters, which are relatively evenly heated along the circumference of the internal surfaces. In the case of a galvanizing tank (despite being a non-pressurized device), this rule is absolutely crucial for the safe operation of the device. This is due to the fact that very intense heat sources are distributed in only a few locations. Temperature unevenness leads to uneven deformation, which results in an uneven stress field. To minimize these unevenness, stops should be made to standardize the gradients.    

The figure below shows a graph of the simulation convergence over time. It shows that the greatest convergence problems occurred in approximately the first 20% of the time. What does this mean? Although the simulation is hypothetical, it does illustrate a possible failure scenario. During the initially uneven heating (i.e., the first hour or two), a deformation state (buckling of the tank's side surface) occurs, which bends the tank's arcs. The arcs could have been at temperatures of several dozen degrees Celsius, but their impact strength was still low enough that the stresses resulting from the deformation led to brittle fracture. Unfortunately, the author has no hard evidence for such a scenario, and never will.

Model of the galvanizing tank used for simulation.

Results for the 5th hour of heating, i.e. for 10% of the entire heating time of the bathtub.

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This section presents simulations for various load combinations of galvanizing bathtub 50 mm thickness

Load case

This section presents simulations for several different load combinations

• Operation temperature +450C

The FEM Model

The galvanizing tank is constructed of 50 mm thick steel sheet, grade S235JR+N / 1.0038, according to the PN – EN 100025 – 2 /1/ standard. The tank model and its support scheme were created based on the drawings below. The zinc brick models were created based on data from a manufacturer of grade Z1 zinc (HCM SHG 99.995). [1] Huty Cynku „Miasteczko Śląskie” S.A.[2]

[1] https://hcm.com.pl/oferta/#cynk-z1

[2] https://hcm.com.pl/

The base model

For this purpose, a basic model of all the essential elements of the bathtub and the zinc brick charge was created, shown below. Depending on the needs, this model will be cut into smaller portions to complete a given simulation batch within a reasonable timeframe, i.e., no longer than one day.

Tools for assessing stress state

Principal stresses were used to assess the stress level. These stresses, designated S1 to S3 in the simulation, have positive or negative values, indicating the nature of the material's work at a given point. Sign convention: positive: the material is in tension; negative: the material is in compression. Bending of the walls also occurs, during which one side of the wall is stretched (positive stress) and the other side is compressed (negative stress). Depending on which side of the tank is viewed (inside or outside) or at which point in the cross-section, different values ​​and signs will be observed.

Principal stresses are perpendicular stresses acting on the walls of an element rotated so that the shear stresses disappear.

S1 (Maximum Principal Stress) is the most "positive" stress. It shows the maximum tension at a given point. Even if S1 is negative, it means the element is compressed from all sides, i.e., a so-called triaxial compression state occurs. Maximum positive values ​​(red/yellow zones) indicate that these areas could fracture under tension. Blue zones are where the "maximum" tension is actually compression (or close to zero).

S3 (Minimum Principal Stress) is dominated by blue (negative values). It shows how strongly the material is being crushed, especially over supports.

S2 (Middle Principal Stress) is useful in shell structures because a flat sheet metal surface typically experiences a so-called biaxial stress state. This means that the material is being pulled or compressed in two directions simultaneously. A good analogy is a stretched drum membrane. One might think that since S2 is "medium," it can be ignored, but this is not the case. S2 is crucial in calculating the equivalent stress. Steel fails through shear (crystal slippage), and shear depends on the difference in stress. If S1 is large and positive, and S2 is large and negative, the equivalent stress will be large. If S1 and S2 are both large and positive, meaning stretching occurs in multiple directions, the equivalent stress will be smaller.

It is common to confuse principal stresses with Huber's reduced stresses.[1]The latter is always positive because it actually reflects the scalar energy of shear displacement. The reduced stress is excellent for assessing the strain in steel, as it indicates the moment when the steel transitions into the plastic state. In the language of linear algebra, a single principal stress is a stress tensor matrix. Looking at the principal stress field, we see a scalar number derived from the tensor field.

[1] Maksymilian Tytus Huber (1872–1950), an outstanding Polish scientist. In 1904, he published the hypothesis of the specific energy of shear deformation.

The results⁰C

The first principal stress, S1, represents the maximum tensile stress. The highest values ​​occur at the upper flange (flange) and at the corners. This is the result of stress concentration at the points of geometry change and the effect of hydrostatic pressure from the zinc, which "pushes" the tank. The value of principal stress S1 suggests that the material has likely exceeded its yield point and is flowing in these areas.

The second principal stress S2 acts perpendicular to S1 and S3. It usually describes the stresses along the plane of the wall [1]Negative values ​​(blue) on the long walls suggest that in certain directions the material is being “squeezed” by thermal expansion constraints from the rigid bottom.

The third principal stress, S3, represents the maximum compressive stress, which in this case primarily represents compressive forces. High compressive stresses can lead to local buckling of the walls if they are too thin, although in this case, this effect is unlikely to occur. However, it can occur in the final phase of the bathtub's service life, when the wall is much thinner than the initial 50 mm. Unfortunately, this combination is not a matter of dispute, as the failure occurred when the bathtub was new.    

[1] In pipeline design, this is the most important circumferential stress

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In AutoPipe, the MSRS method is easy to implement. First, you need to group the system segments as needed. Let's say you're using the same model divided into three sections.

We select the segment and group using a dedicated command. This allows us to assign a different SAM combination to each group.

It is very easy to define combinations of seismic sects in very different ways

The results are in a very accessible form: For the MSRS 1 combination:

MARS 2 looks different because the combination of spectra was different.

Caesar, unfortunately, is two classes worse. It doesn't have a convenient menu for dividing into zones. We define dynamic forcing combinations. We can enter a range of nodes, but it's not required. If left blank, it means the entire system will be considered.

The results can only be viewed in table format, which is disappointing compared to AutoPipe. For MSRS 1, it looks like this:

For MSRS 2 is:

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In B31.3 it is also presented in this way, except that there is no mention of the movement of supports during an earthquake.

EN 13480-3, however, uses a more lenient statement (should). Such analyses are typically described with the acronym SAM – Seimic Anchor Movement, although they apply not only to fixed points but also to supports with line guise and line stop functions. Of course, they do not apply to rest functions.

First, I'll return to the SAM static analysis, which is the most common approach. The decision to use statics or dynamics is left to the designer. The model looks as follows: The entire structure is divided into three SIB (Seismic Interface Barrier) segments, the horizontal section being the viaduct, and the vertical section being the approach to the building.

Pipe specification: DN 100 (4'), 6.02mm (STD). The medium is water. A horizontal seismic acceleration of 0.2g and a vertical seismic acceleration of 0.1g was assumed. The vertical section height was 8m. The temperature was 100C and the pressure was 1 MPa. The seismic displacement was determined according to applicable rules, which I will not describe now: NSEW overpass level = 5mm, NSEW building entrance level = 40mm. No seismic rotation at the supports was assumed.

To verify that the correct directions have been chosen, it's worth viewing the displacement results. If all is well, the SIBs will change their displacement direction.

The result of the static SAM analysis in AutoPipe for the fixed point upstream of SIB 1 is as follows. The code stress is 40.8 MPa.

Regarding Caesar, the situation is less convenient. All seismically movable supports must be equipped with Cnodes. For the same point, the stress was 47.6 MPa.

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In Caesar II the model looks like this:

Both models, of course, received identical seismic excitations. The accelerations in each direction are slightly different. They act on the ENTIRE model simultaneously. The acceleration unit is mm/s², meaning 80 mm/s² is just 0.08 m/s², which is very low. There are no defined seismic displacements. For simplicity, wind and snow were omitted. Therefore, the only occasional load is seismic.

The results for the static analysis in AutoPipe are typical and not cause for concern. The stress on the arc ranges from 29 to 45%.

After adding the dynamic analysis, the result appears with the designation R1. This is the Response Spectra result No. 1. The effort is only 3 to 4% for SUS+R1.

Static results on Caesar II are similar. The strain for the same load combination is 33%.

Unfortunately, Caesar II's dynamic analysis results leave much to be desired. First, they're not presented graphically, only numerically. Therefore, we have to exit the dynamics section and open the Input section to see where the node is at its peak intensity. THIS IS TERRIBLE SOLUSION..

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AutoPipe Advanced 2024 has the following standards available for spectral generation. It includes IBC, but surprisingly, it doesn't include ASCE7. Instead, it does have the basic European Eurocode standard and a strange Spanish standard.

Caesar II, however, has a slightly different list. It primarily includes ASCE 7 and IBC, but no European standards. Perhaps it's a sign of the times...

In this situation, the only thing that could be compared are the generation results according to IBC. AutoPipe Advanced 2024 only has the 2006 version, which is somewhat puzzling. However, Caesar II v14 has the 2021, 2018, 2012, 2006, and 2000 versions available. This shows that I can only use the 2006 version for comparison purposes.

In Autopipe Advanced 2024, you can generate a seismic spectrum for any point in the US. Simply enter the longitude and latitude (I entered data for the Midwest here), and the four data points needed to generate the spectrum are automatically populated. There's even a zip code option (!), but I haven't used it.

There is also a very useful option to refine the mesh by using the variable below:

As a result, we get this very nice graph:

Caesar II v14 has a slightly different menu. Note the Importance Factor, which isn't included in this standard. Its values ​​depend on the type of building. I filled in the remaining factor values ​​with the same values ​​as those generated by the AutoPipe generator. In Caesar II, they must be entered manually, which is very tedious and can also generate errors.

The second factor not included in AutoPipe is Response Modification R, which is included in the 2024 IBC version. I don't have the 2006 version, so it's hard for me to say why AutoPipe Advanced 2024 doesn't include this factor.

Caesar II generates a graph but with much less density.

The huge difference in density and the inability to adjust it in Caesar II make the data range for spectrum plotting incomparable. Unfortunately, there's a significant drawback to Caesar II.

Comparison of acceleration results for several points gives satisfactory results.

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The model is identical for both programs: 10′, STD, A106B, air 0.5 MPa, +100C. Length 15m. The natural frequency for a restrained beam is described by the well-known formula given below. For deformation mode I, f = 1.18 Hz

The results in AutoPipe for mode I are 1.16 Hz when dividing the pipe into 10 equal parts

The results in AutoPipe for mode I are 1.13 Hz without division.

The results in CII for mode I are 1.17 Hz when dividing the tube into 10 equal parts and using a lumped mass model.

The CII results for mode I are 0.82 Hz without splitting and with a lumped mass model.

However, for the consistent mass model, the results are much better. It's true that the analysis takes longer, but for simple models, this doesn't matter.

This simple comparison shows that without dividing long sections into relatively short ones, Caesar II, and using a simplified mass model (lumped), produces results that differ significantly from those described by the formula, but also from those of AutoPipe. Furthermore, AutoPipe provides a very nice figure for higher strain modes (see below), which CII does not.

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DLF determined in accordance with the harmonized standard is 2.0. It can be seen that for PSV valves, which generally operate very quickly, DFL = 2.0 can be used by default.

At this point, the x DLF force can be used in static analysis, but also in dynamic analysis. Static analysis is quite straightforward, so I turned to dynamics.

First, I generate a flat spectral DLF = 2 with the valve opening time calculated above of 1.304 ms.

I add the thrust force from the manufacturer's data sheet and then the load combination.

The results are interesting.

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