Thermal/pressure Loaded Cylinders

Chapter 12: Thermal/Pressure Loaded Cylinders

12

Thermal/Pressure Loaded Cylinders 

Summary



Introduction



Required Solutions



FEM Solutions



Results



General Analysis Tips



Input File(s)

212 213 213

213

216

219

219

212 MD Demonstration Problems CHAPTER 12

Summary Title

Chapter 12: Thermal/Pressure Loaded Cylinders

Contact features

• Curved contact surfaces • Deformable-deformable contact

Geometry and description

Two eccentric cylinders: t = 0.03 “ R = 0.32 “ R = 0.25 “

0.09 “

Material properties

Inner cylinder: Isotropic elasto-plastic; E inner = 2.2 107 psi ;  in = 0.3 , –5

Thermal expansion coefficient = 1.85 10 1  F ,

Initial yielding stress: 9900 Psi; Piece-wise

linear and isotropic work hardening rule. Outer cylinder: Isotropic elastic, Young’s modulus is temperature dependent, initial value E outer = 1.27 107 psi ;  out = 0.3 , Thermal expansion coefficient = 1.85 10–5  F , no plasticity. Analysis type

Quasi-static analysis; Material nonlinearity (softening by temperature and hardening by plastic deformation); Geometric nonlinearity

Displacement Boundary conditions and applied loads

Symmetric displacement constraint over the horizontal plane with one end of the cylinders are fixed in the z-direction. Step 1: Thermal loading 50oF temperature change. Step 2: Internal pressure loading; internal cylinder.

Element type

8-node linear elements

Contact properties

Deformable-to-deformable body contact without friction

FE results

Plot of stress/strain and displacement distribution after each step.

Displacement Contours after Step 2

CHAPTER 12 213 Thermal/Pressure Loaded Cylinders

Introduction This application example evaluates the performance of an adaptive load stepping scheme in the applications of MD. Nastran SOL 400 for the FE analysis. Due to the symmetry condition, half of the assembly is sufficient for the finite element analysis. This example involves thermal load, contact, material, and geometrical nonlinearity under pressure loading. The geometry and material descriptions are given in the above summary table. There are two load steps. The first step is to apply the thermal load by specifying the temperature changes at each node of the two eccentric cylinders. With the thermal loading along with the given boundary conditions, the stress and strain are generated due to uneven thermal expansion of the two cylinders. In the second loading step, a pressure is applied at the inside of the inner cylindrical surface. Due to this pressure, the smaller cylinder expands in diameter and eventually fills the gap between the two cylinders when the outer surface of the small cylinder progressively touches the inner surface of the outside cylinder. Due to the strong nonlinearity, adaptive time stepping scheme is used. By the adaptive time stepping scheme, the step size of each increment is adjusted at the end of step that just converged.

Required Solutions SOL 400 is used for the FE analysis of this problem. The advanced HEX element defined by PSOLID entry pointing to an auxiliary PSLDN1 entry is used. For the first loading step, the thermal strains and stresses of the two cylinders are of the interests. For the second load step, the deformation and contact between two cylinders under pressure loading are investigated. Due to the nonlinearity introduced by nonlinear material properties and contact, convergence speed varies with the nonlinear deformation and changes of contact condition. In order to achieve fast and stable analysis, the time step size is automatically adjusted according to the convergence condition. In the current version of MD Nastran SOL 400, this is done by adding the NLAUTO option into the input data file. For comparison purposes, one analysis with MSC.Marc version 2005 with the solid element of the same formulation as the element in SOL 400 and auto step scheme is also conducted.

FEM Solutions The element, contact, material/geometry, solution algorithm, and convergence schemes parameters are explained in this chapter.

The Advanced HEX Element The FE model is shown in Figure 12-1. As mentioned earlier, two solutions are obtained. The first solution was obtained by using the MD Nastran SOL 400 with the advanced HEX element, which is defined by the PSOLID and PSLDN1 bulk data options as shown below, where (C8 SOLI L) defines the 3-D continuum solid element with linear integration scheme. PSOLID PSLDN1 +

1 1 C8

1 1 SOLI

0 + L

214 MD Demonstration Problems CHAPTER 12

Figure 12-1

The FE Model for the Numerical Solution

Contact Parameters As shown in Figure 12-1, the contact body named as cbody1 (shown in pink) represents the inner cylinder. The contact body named as cbody2 defines the outside cylinder. The black arrows represent the pressure applied on the inner surface of the small cylinder (cbody1). It should be noted that only half of the whole assembly is modeled due to the symmetry condition. In the input data file, the contact bodies are defined deformable contact bodies as below: BCBODY 1 BSURF

1

1813

BCBODY 2 BSURF

2

3D

3D 1013

DEFORM 1 1814

1815 1816

DEFORM 2 1014

0 1817

1818

1819

1017

1018

1019

0

1015 1016

The BCTABLE bulk data entries shown below define the touch conditions between the bodies: BCTABLE

BCTABLE

BCTABLE

0 SLAVE

1 0 FBSH MASTERS 2 1 SLAVE 1 0 FBSH MASTERS 2 2 SLAVE 1 0 FBSH MASTERS 2

0. 0 1.+20

1 0. 0 0.

0. 0 1.+20

1 0. 0 0.

0. 0 1.+20

1 0. 0 0.

0.

0.

0

0.

0

0.

0

0. 0. 0. 0. 0.

As shown above, BCTABLE with ID 0 is used to define the touch conditions at the start of the analysis. 0 identifies the case number. This BCTABLE is mandatory for the contact analysis with SOL 400. Also, the options BCONTACT with

CHAPTER 12 215 Thermal/Pressure Loaded Cylinders

ID 0 and BCPARA with ID 0 are all applied at the start of the analysis. For each load step, the touch condition can be defined by BCTABLE, BCPARA, and BCONTACT option.

Material/Geometry Parameters Both bodies in this analysis are isotropic in terms of thermal and mechanical properties. Body one represents the inner cylinder, which is also elasto-plastic. The Young’s modulus, Poisson ratio, and thermal expansion coefficient are defined by MAT1 bulk data option. The plasticity properties are defined by MATEP with TABLES1 option. Here, TABLES1 is associated with MATEP to defined the strain hardening rule of the material with ID 1. MATEP MAT1 TABLES1

1 1 1 0. .00615

Table 2.2+7 2 9900. 20000.

3.9-4 .05

1 .3

1.

Isotrop Addmean 1.85-5

12500. 25000.

9.5-4 .1

15200. 28000.

.00295 ENDT

17500.

Body two represents the outside cylinder. As shown below, this body has a temperature dependent Young’s modulus (see TABLEM1). MAT1 MATT1 TABLEM1

2 2 2 0.

2.2+7 2 2.2+7

50.

.3

1.

1.85-5

1.76+7

100.

1.54+7

ENDT

The thermal expansion coefficient of the two cylinder are the same which is 0.0000185 1/oF.

Case Control Parameters There are two loading sequences (or loading steps) in the analysis. In each loading sequence, the control parameters are defined by the NLPARM and the NLAUTO option. The ID of the NLAUTO option is linked with the identification number of the NLPARM option. This option must be used in conjunction with NLPARM. The NLAUTO options are specified in the bulk data area. As shown below, load STEP ID 1 of SUBCASE ID 1 defines all necessary conditions applied to the analysis for the first load step which includes bulk data options (TITLE, NLPARM, BCONTACT, SPC, LOAD) and the requested output information. Particularly, it is necessary to note the analysis control options of NLMOPTS and the LGDISP parameter. In this example, the NLMOPTS option defines LRGS to 1. It means that LARGE STRAIN formulation is used. The LGDISP parameter indicates that geometric nonlinearity includes the stiffness of follower forces. NLPARM defines the parameters to control the time step and convergence schemes. In this example, PFNT means that full Newton-Raphson method is adopted. The attempted total number of loading increments is set to 20. The maximum iteration for each increment is set to 25. UP means the convergence scheme is set to check both the convergence of displacements and residuals. In this loading sequence, both tolerances are set as 0.01. It is worth to note that a negative value is set for the displacement check. The negative sign means the convergence check will be based on the incremental displacement. And NO in the NLPARM option means that it is not required to output the analysis results for intermediate loading steps, except the results at the end of the loading sequence. However, the total number of loading increment may be changed according to the parameters set in NLAUTO option. In the first load step, the deformation is relatively small. The desired number of iterations (1st field of the second line of NLAUTO option) is set as 5. In the second load step, due to contact and large deformation, the desired number of iteration is set as 7. To set a proper desired number of iterations is critical to achieve the solution with minimum computation time and adequate

216 MD Demonstration Problems CHAPTER 12

accuracy. Too large numbers may cause significant change of time step size between increments, which may cause the solution to converge slowly or even diverging. If this happens, SOL 400 cuts the time step size back. As one of the consequences, the analysis may need even longer computation time. To avoid this, it is recommended to set a reasonably small value for the maximum ratio of incremental step size change between incremental steps (the 6th field of the first line of the NLAUTO option). This parameter is set as 10 with desired number of iteration as 5 for load step 1. For the second load step, with consideration of the fact that contact and large deformation may occur, this parameter is set as 1.2 with desired number of iterations as 7. This is particularly important in order to avoid penetration and also to control the time step size with good balance of efficiency and accuracy. SUBCASE 1 STEP 1 TITLE=This is a default subcase. ANALYSIS = NLSTATICS NLPARM = 1 BCONTACT = 1 SPC = 2 LOAD = 3 TEMPERATURE(LOAD) = 4 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL STRESS(SORT1,REAL,VONMISES,BILIN)=ALL BOUTPUT (PRINT)=ALL STEP 2 TITLE=This is a default subcase. ANALYSIS = NLSTATICS NLPARM = 2 BCONTACT = 2 SPC = 2 LOAD = 6 TEMPERATURE(LOAD) = 8 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL STRESS(SORT1,REAL,VONMISES,BILIN)=ALL $ Direct Text Input for this Subcase BEGIN BULK NLMOPTS LRGS 1 PARAM LGDISP 1 NLPARM 1 20 PFNT 1 -0.01 0.01 0 NLAUTO 1 0.05 1.0 0.1 10. 5 1 0 0 10 $ NLAUTO 2 0.05 1.0 0.1 1.2 7 1 0 0 10 NLPARM 2 20 PFNT 1 -0.01 0.01

25 UP 0 1.0e-5 0.2 0 0

999999 0.0

1.0e-5 0 25

999999 0.0 NO

0.2 0 UP

NO

Results Load Step One The initial temperature of the whole assembly is set as zero (0). In the first load step, a temperature load is applied to the inner cylinder and part of the outside cylinder (see Figure 12-2 - yellow color). Due to the thermal expansion caused by the temperature load and the corresponding changes of the material properties, thermal strain and stress are generated. Figure 12-3 shows the distribution of major principal stress and the equivalent stress at the end of this load step. It is seen that the distribution of stress is uneven through the wall thickness of the outside cylinder. However, the stress in the inner cylinder is quite uniformly distributed (see Figure 12-3(b)). This is because the inner cylinder has a

CHAPTER 12 217 Thermal/Pressure Loaded Cylinders

uniform temperature load with minimum displacement boundary constraints. Therefore, it has nearly stress-free thermal expansion. With the adaptive loading step scheme, the analysis of this loading sequence is completed in eight incremental steps.

Figure 12-2

Temperature Loading

(a) Major Principal Thermal Stress

Figure 12-3

(b) Equivalent Stress

Distributions

Load Step Two This load step is to apply the pressure inside the inner cylinder. Due to the pressure loading, the inner cylinder expands in diameter. At some point of loading, the gap between the two cylinders is closed. Figure 12-4 (a) shows the gap between two cylinders at the beginning of this load step. Figure 12-4 (b) shows that the gap is completely closed after the pressure is fully applied. Using the adaptive load step control, this load step is completed in 19 incremental steps. So the total number of incremental steps for the analysis is 27 steps. The distribution of equivalent stress in the deformed cylinders is shown in Figure 12-5. It is seen that the level of stress is higher in the inner cylinder. The lowest stress occurs on the outside cylinder along its inner surface which is in contact with the outside surface of

218 MD Demonstration Problems CHAPTER 12

the inner cylinder. The lower level of stress is mainly because of the softening of material due to increased temperature.

(a)

(b)

Figure 12-4

Change of Contact Status Between the Two Cylinders

Figure 12-5

Equivalent Stress of the Deformed Cylinders After Pressure Loading

In addition to the analysis with MD Nastran SOL 400, the MSC.Marc 2005 version is also used to conduct the analysis with the same type of element and material and boundary condition definition. The results are quite close as shown in Figure 12-6(a) and Figure 12-6(b). The analysis by MSC.Marc takes 16 incremental steps for the first load step and another 27 incremental steps for the pressure loading step.

CHAPTER 12 219 Thermal/Pressure Loaded Cylinders

(a) MD Nastran SOL 400

Figure 12-6

(b) MSC.Marc

Displacement Contours of the Cylinders After Pressure Loading

General Analysis Tips Convergence control: While the nonlinearity is quite strong in the second load step, it is suggested to use both displacement and residual convergence check due to the nonlinearity introduced by contact. Also, the full NewtonRaphson iteration scheme is recommended for all SOL 400 analyses because the degree of nonlinearity is typically significant. Adaptive step size control: The NLAUTO option with NLPARM option provides the convenient interface for user to control the analysis procedure. Proper setting of the control parameters is very important to obtain accurate results without losing computational efficiency. A useful tip is to use loose control over the desired number of iteration but use tighter control over the maximum ratio of time step change allowed after each converged step. Contact control: In this example, the FE nodes of inner cylinder part are defined as slave contact nodes. This is due to the consideration that, during the pressure loading process, the inner cylinder will expand and intend to touch the inner surface of the outside cylinder. In this case, the nodes on the inner cylinder surface usually have much larger incremental displacements at each increment.

Input File(s) File

Description

nug_12bm.dat

Input data for MD Nastran SOL 400

mdug_12b3d.dat

Input data for Marc