In-depth studies61 - Newsletter EnginSoft Year 12 n°2 In-depth studies Newsletter EnginSoft Year 12 n°2 - 60

one end to the other, there must be a failure in the system boundary, i.e. there’s a leak! As well as ensuring that all the mass that enters the system leaves at the desired point, it may also be that thermal energy must be conserved: thus insulation and fluid/wall temperatures can also form a critical part of the network simulation.From the perspective of the fluid, at its most basic the system consists of a series of connected components which are designed to variously:• Store a volume of fluid (e.g. reservoirs or accumulators)• Add or remove energy from the fluid (e.g. pumps, compressors, heat

exchangers)• Direct the kinetic energy of the fluid (e.g. pipes, bends, junctions)• Control the flow rate of the fluid (e.g. valves, transitions, orifice

plates)There are of course many other component types than those listed here, but these basic categories will serve to illustrate the principles for now.If it were possible to find a way to characterise the thermo-fluid effect of these components in a way that could be made generally applicable, system simulation would not have to rely on the discretization approach described above. Happily, this is possible by means of system level thermo-fluid analysis. Although there is a slight complication in that not all fluids are created equal.

Thermodynamics and Fluid DynamicsOur everyday experience of fluids makes it clear that they possess very different characteristics. Walking through fresh air on the way to work is a very different experience from wading across a swimming pool; getting tomato ketchup out of a bottle is a very different experience than the same operation with water. Even the same fluid can look and behave very differently at different pressures and temperatures. For instance, a boiling kettle contains water in two distinct phases as well as in an intermediate – two-phase – condition.Effectively accounting for the changes in fluid properties and energy content throughout a system is an important but complex task. Accordingly, some thought has to go in to the level of detail that’s appropriate to achieve the correct balance between useful accuracy and solution complexity. For the user, complexity can manifest itself in terms of both excessive and intricate input demands and also solution times that extend into hours, rather

than seconds or minutes. Balancing the conflicting requirements is a delicate task and as might be expected, there is no one-size-fits-all solution. For example, air circulating through the ventilation network of an aircraft’s cabin may be reasonably modelled using ideal gas assumptions, but the same may not be true of a high pressure multicomponent gas line. One option may be to provide the user with the option of perfect, ideal or real gas models. Two-phase and Non-Newtonian fluids can prove more complex again and the accuracy of any simulation will depend strongly upon getting accurate data for the exact fluid under consideration.The physics of phase-change is another extremely complex field. Pipe orientation, fluid velocity and the rate of heat addition will all affect the behaviour of the fluid. Perhaps unsurprisingly given the apparent sensitivity of the physics of phase change to external factors, experimentalists have derived numerous correlations to account for heat transfer to and from

two-phase fluids. Which model is most appropriate for a given situation will be something that must be judged by the user.

In the second part of this article, we will examine in depth the topic of System Level Thermo-Fluid Analysis speaking about pressure drops in a system and about time.

This article is a reprint of the original published by Mentor Graphics: http://bit.ly/YC5Qec

New Webinar: What is System Level Thermo-Fluid Analysis?Watch the podcast on EnginSoft channel: https://vimeo.com/user18735961

For more information:Alberto Deponti, EnginSofta.deponti@enginsoft.it

Figure 3 - Air density as a function of temperature and pressure. In Flowmaster, fluid properties can be defined based on NIST REFPROP database. Accurate data for fluid properties are crucial for valuable simulation results

In ANSYS Workbench there are three methods to determine the buckling load of a structure:• Linear based eigenvalue buckling analysis• Nonlinear based eigenvalue buckling analysis (new in R16)• Nonlinear buckling analysis

Linear based eigenvalue analysisAn eigenvalue buckling analysis allows the calculation of the theoretical bucking load of an ideal elastic structure (without any imperfection). In real world applications, geometric imperfections and nonlinear behaviour prevent a structure from achieving its theoretical bucking load (fig.1). This kind of analysis allows results to be obtained quickly, but linear buckling generally yields un-conservative results: the buckling load calculated with this method is usually superior to the real buckling load.

The first step in performing an eigenvalue buckling analysis is to introduce a linear elastic, pre-buckling state into the structure under investigation (the load direction has to generate instability in the structure). In such a pre-buckling state, the load generates a stress state {s}. The stress state {s} will modify the structural stiffness: such effect (Stress Stiffening effect), presents itself through the Stress Stiffness Matrix [S], which is summed to the structure stiffness matrix. When instability occurs (that is the achievement of the critical load), the structure can undergo great displacements, even for small load increments. With such assumptions, it is possible to get the eigenvalue equation for the linear based buckling analysis:

([K]+λi [S]){ψi }=0

Where: λi: i-th load multiplier (i-th eigenvalue) Ψi, i-th buckling mode.

Once the load multiplier is known, the buckling load can be calculated as:

Fbuckling=λ*Fapplied

In ANSYS Workbench, the eigenvalue buckling analysis is set by performing a structural static analysis first to account for the pre-buckling state, and then by linking the Solution branch to the Setup branch of the Eigenvalue Buckling analysis block (fig.2).

Nonlinear based eigenvalue buckling analysis (new in R16)When performing a linear buckling analysis, the static analysis has to be linear and therefore should not contain geometric, material or contact nonlinearities. With the non-linear based approach, already included in ANSYS APDL and available in Workbench starting from R16, it’s possible to take nonlinearities into account even in the pre-buckling phase, which is in the earlier static analysis.What’s the difference between the linear-based buckling analysis approach and the nonlinear based buckling analysis one? In both cases the aim is to solve an eigenvalue problem, but the nonlinear-based methodology allows to take into account the variation of the stiffness matrix, due to nonlinearities, during the pre-buckling analysis. The equation solving the eigenvalue problem is now:

([KT ]+λi [S]){ψi }=0

Where [KT] is the tangent stiffness matrix (fig.3) calculated at the last time of the pre-buckling static analysis.

When a nonlinear based eigenvalue analysis is set, several aspects have to be taken into account:• In order to activate this computation methodology, at least one

nonlinearity (related to material, geometry or contact) has to be defined in the pre-buckling static analysis;

• Besides the loads defined in the static analysis, it’s also necessary to define at least one load in the buckling analysis, so to proceed with the solution. In order to manage this aspect, the option “Keep Pre-Stress Load Pattern” (fig. 4) has been introduced: setting this property to “Yes”, the same load, already defined in the static analysis, will be applied in the buckling analysis; setting this property to “No”, it will be possible to define new loads in the buckling analysis, different from those applied to the static one (in this release the loads definable in the buckling analysis are: Thermal Condition, Nodal Force, Nodal Pressure, Nodal Displacement)

Buckling analysis in ANSYS

Figure 1

Figure 2

Figure 3

Figure 2 - Particular of a the model of a cooling system implemented in Flowmaster, the system level thermo-fluid analysis software that allows to easily and quickly model complex systems. This model was used to perform a steady analysis for balancing all the branches of the cooling system and a transient analysis for studying water hammer phenomena

In-depth studies63 - Newsletter EnginSoft Year 12 n°2 In-depth studies Newsletter EnginSoft Year 12 n°2 - 62

Determination of the buckling pressure with nonlinear based eigenvalue analysisIn order to apply the nonlinear based computation methodology, geometric nonlinearities have been activated and a material plastic behavior (bilinear isotropic hardening) has been defined. The buckling, nonlinear based analysis provides a critical load multiplier equal to 0.37655. Therefore the buckling pressure is equal to:

pbuckling=0.1+0.1*0.37655=0.1377 MPa

The “Keep Pre-Stress Load Pattern” option was set to YES, therefore the applied pressure in the buckling analysis is the same as the one applied in the static analysis (0.1 MPa).It can be noticed that the value of the buckling pressure achieved with the nonlinear based approach is lower than the value obtained with the linear analysis, and therefore much closer to the reference value.

Determination of the buckling pressure with nonlinear static analysis Using the upgeom operation, a perturbed mesh has been created, which is based on the first ten buckling modes (calculated with the linear buckling analysis), each one of them scaled by a 0.1 factor. The small implemented APDL macro is presented here:

/prep7s_factor=0.1*do,i,1,10upgeom,s_factor,1,i,file,rst ! Add imperfections as a tenth of each mode shapecdwrite,db,cylinder_perturbed,cdb*enddo

The macro is loaded into a Mechanical APDL block and linked to the solution branch of the Eigenvalue Buckling analysis. It is then possible to recover the cdb file containing the perturbed mesh from the WB project folder: the mesh can then be used for the nonlinear calculation, by importing it through an external model block.

It’s necessary to apply a load equal or superior to the buckling load to induce structural instability. A pressure of 0.19 MPa (a value superior to the one calculated with the linear buckling analysis) is then applied to the perturbed model and a nonlinear analysis is carried out (including also in this case the geometric and material nonlinearities). It’s necessary to set a sufficiently high number of sub-steps to catch the point in the analysis where the stiffness matrix becomes singular (i.e. when the buckling load has been reached).

In the case under investigation, the analysis begins to have convergence troubles at the time = 0.64125. Therefore the buckling pressure becomes:

pbuckling=0.19*0.64125=0.1218 MPa

ConclusionsThe following table presents a synthesis of the results that have been achieved using the three methods (Table 1). It can be noticed that the most accurate result is the one provided by the nonlinear analysis. Nevertheless the new Nonlinear based eigenvalue analysis allows to achieve a result much closer to reality in comparison with the linear method, because it can account for the nonlinearities in the pre-buckling phase.

Alessandro Preda, EnginSoft

The load multipliers only scale the loads applied in the buckling analysis; the buckling load is determined as follows:

Fbuckling=Frestart+λ*Fperturbed

Where:• Frestart is the load applied in the static analysis (pre-buckling)• Fperturbed = Frestart if “Keep Pre-Stress Load Pattern” = Yes• Fperturbed = load defined by the user in the buckling analysis if “Keep

Pre-Stress Load Pattern” = No

Nonlinear buckling analysisThe buckling load can also be determined by performing a nonlinear static analysis: this approach is much heavier computationally, but it provides a more realistic result.In order to determine the buckling load, by performing a nonlinear static analysis, the following procedure has to be followed:1. Linear based eigenvalue analysis: identify the buckling modes;2. Imperfections generation: use the buckling modes calculated

with the linear analysis to generate the geometric imperfections (upgeom operation), which are necessary to trigger the buckling in the structure. The imperfections have to be in the same order of magnitude as the manufacturing tolerances of the structure;

3. Launch of the nonlinear analysis: when the instability point is reached, the tangent stiffness matrix becomes singular; the Newton Raphson method will start having convergence difficulties: in this way the buckling load can be identified.

Application caseIn order to compare the results obtained from the three different methods presented above, an example can be taken into account where a cylindrical structure undergoes an external uniform pressure. In particular, we can consider a circular cylinder made of 2024-T3 aluminum alloy which is internally stiffened by 5 rings with a Z section, and which undergoes a uniform external pressure of 0.1 MPa. The cylinder is constrained in 3 nodes, in order to prevent all rigid translations and rotations.

The reference value of the buckling pressure for the specific cylinder is 0.121 MPa: this value comes from an experimental test performed on the real cylinder (ref: Dow, Donaldson A., November, 1965, “Buckling and Post-buckling Tests of Ring-Stiffened Circular Cylinders Loaded by Uniform External Pressure.”, NASA Technical Note NASA TN D-3111, Langley Research Center).

Determination of the buckling pressure with linear based eigenvalue analysisThe linear based buckling analysis provides a critical load multiplier (that is the lowest among the calculated ones) equal to 1.4222. Therefore the buckling pressure is equal to:

pbuckling=0.1*1.4222=0.1422 MPa

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8 - Linear based eigenvalue analysis: first buckling mode shape

Figure 9

Figure 10 - Example of perturbed mesh generated with upgeom

Table 1