Abaqus Example Problems Manual
Abaqus 6.11
Example Problems Manual
Abaqus
Example Problems Manual
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Abaqus Version 6.11 ID: Printed on:
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Abaqus Version 6.11 ID: Printed on:
CONTENTS
Contents Volume I
1. Static Stress/Displacement Analyses Static and quasi-static stress analyses
Axisymmetric analysis of bolted pipe flange connections 1.1.1 Elastic-plastic collapse of a thin-walled elbow under in-plane bending and internal
pressure 1.1.2
Parametric study of a linear elastic pipeline under in-plane bending 1.1.3 Indentation of an elastomeric foam specimen with a hemispherical punch 1.1.4
Collapse of a concrete slab 1.1.5
Jointed rock slope stability 1.1.6
Notched beam under cyclic loading 1.1.7
Uniaxial ratchetting under tension and compression 1.1.8 Hydrostatic fluid elements: modeling an airspring 1.1.9 Shell-to-solid submodeling and shell-to-solid coupling of a pipe joint 1.1.10
Stress-free element reactivation 1.1.11
Transient loading of a viscoelastic bushing 1.1.12
Indentation of a thick plate 1.1.13
Damage and failure of a laminated composite plate 1.1.14
Analysis of an automotive boot seal 1.1.15
Pressure penetration analysis of an air duct kiss seal 1.1.16 Self-contact in rubber/foam components: jounce bumper 1.1.17 Self-contact in rubber/foam components: rubber gasket 1.1.18 Submodeling of a stacked sheet metal assembly 1.1.19 Axisymmetric analysis of a threaded connection 1.1.20 Direct cyclic analysis of a cylinder head under cyclic thermal-mechanical loadings 1.1.21 Erosion of material (sand production) in an oil wellbore 1.1.22 Submodel stress analysis of pressure vessel closure hardware 1.1.23 Using a composite layup to model a yacht hull 1.1.24 Buckling and collapse analyses
Snap-through buckling analysis of circular arches 1.2.1 Laminated composite shells: buckling of a cylindrical panel with a circular hole 1.2.2
Buckling of a column with spot welds 1.2.3
Elastic-plastic K-frame structure 1.2.4
Unstable static problem: reinforced plate under compressive loads 1.2.5 Buckling of an imperfection-sensitive cylindrical shell 1.2.6
CONTENTS
Forming analyses
Upsetting of a cylindrical billet: quasi-static analysis with mesh-to-mesh solution
mapping (Abaqus/Standard) and adaptive meshing (Abaqus/Explicit) 1.3.1
Superplastic forming of a rectangular box 1.3.2
Stretching of a thin sheet with a hemispherical punch 1.3.3
Deep drawing of a cylindrical cup 1.3.4
Extrusion of a cylindrical metal bar with frictional heat generation 1.3.5
Rolling of thick plates 1.3.6
Axisymmetric forming of a circular cup 1.3.7
Cup/trough forming 1.3.8
Forging with sinusoidal dies 1.3.9
Forging with multiple complex dies 1.3.10
Flat rolling: transient and steady-state 1.3.11
Section rolling 1.3.12
Ring rolling 1.3.13
Axisymmetric extrusion: transient and steady-state 1.3.14
Two-step forming simulation 1.3.15
Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic
analysis 1.3.16
Unstable static problem: thermal forming of a metal sheet 1.3.17 Inertia welding simulation using Abaqus/Standard and Abaqus/CAE 1.3.18 Fracture and damage
A plate with a part-through crack: elastic line spring modeling 1.4.1 Contour integrals for a conical crack in a linear elastic infinite half space 1.4.2 Elastic-plastic line spring modeling of a finite length cylinder with a part-through axial
flaw 1.4.3
Crack growth in a three-point bend specimen 1.4.4
Analysis of skin-stiffener debonding under tension 1.4.5 Failure of blunt notched fiber metal laminates 1.4.6 Debonding behavior of a double cantilever beam 1.4.7 Debonding behavior of a single leg bending specimen 1.4.8 Postbuckling and growth of delaminations in composite panels 1.4.9 Import analyses
Springback of two-dimensional draw bending 1.5.1
Deep drawing of a square box 1.5.2
2. Dynamic Stress/Displacement Analyses Dynamic stress analyses
Nonlinear dynamic analysis of a structure with local inelastic collapse 2.1.1
Detroit Edison pipe whip experiment 2.1.2
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CONTENTS
Rigid projectile impacting eroding plate 2.1.3
Eroding projectile impacting eroding plate 2.1.4
Tennis racket and ball 2.1.5
Pressurized fuel tank with variable shell thickness 2.1.6
Modeling of an automobile suspension 2.1.7
Explosive pipe closure 2.1.8
Knee bolster impact with general contact 2.1.9
Crimp forming with general contact 2.1.10
Collapse of a stack of blocks with general contact 2.1.11
Cask drop with foam impact limiter 2.1.12
Oblique impact of a copper rod 2.1.13
Water sloshing in a baffled tank 2.1.14
Seismic analysis of a concrete gravity dam 2.1.15
Progressive failure analysis of thin-wall aluminum extrusion under quasi-static and
dynamic loads 2.1.16
Impact analysis of a pawl-ratchet device 2.1.17
High-velocity impact of a ceramic target 2.1.18
Mode-based dynamic analyses
Analysis of a rotating fan using substructures and cyclic symmetry 2.2.1 Linear analysis of the Indian Point reactor feedwater line 2.2.2 Response spectra of a three-dimensional frame building 2.2.3
Brake squeal analysis 2.2.4
Dynamic analysis of antenna structure utilizing residual modes 2.2.5 Steady-state dynamic analysis of a vehicle body-in-white model 2.2.6 Eulerian analyses
Rivet forming 2.3.1
Impact of a water-filled bottle 2.3.2
Co-simulation analyses
Closure of an air-filled door seal 2.4.1
CONTENTS
Volume II
3. Tire and Vehicle Analyses Tire analyses
Symmetric results transfer for a static tire analysis 3.1.1
Steady-state rolling analysis of a tire 3.1.2
Subspace-based steady-state dynamic tire analysis 3.1.3 Steady-state dynamic analysis of a tire substructure 3.1.4 Coupled acoustic-structural analysis of a tire filled with air 3.1.5
Import of a steady-state rolling tire 3.1.6
Analysis of a solid disc with Mullins effect and permanent set 3.1.7 Tread wear simulation using adaptive meshing in Abaqus/Standard 3.1.8 Dynamic analysis of an air-filled tire with rolling transport effects 3.1.9
Acoustics in a circular duct with flow 3.1.10
Vehicle analyses
Inertia relief in a pick-up truck 3.2.1
Substructure analysis of a pick-up truck model 3.2.2 Display body analysis of a pick-up truck model 3.2.3
Continuum modeling of automotive spot welds 3.2.4
Occupant safety analyses
Seat belt analysis of a simplified crash dummy 3.3.1
Side curtain airbag impactor test 3.3.2
4. Mechanism Analyses
Resolving overconstraints in a multi-body mechanism model 4.1.1
Crank mechanism 4.1.2 Snubber-arm mechanism 4.1.3 Flap mechanism 4.1.4 Tail-skid mechanism 4.1.5 Cylinder-cam mechanism 4.1.6 Driveshaft mechanism 4.1.7 Geneva mechanism 4.1.8
Trailing edge flap mechanism 4.1.9
Substructure analysis of a one-piston engine model 4.1.10 Application of bushing connectors in the analysis of a three-point linkage 4.1.11
Gear assemblies 4.1.12
5. Heat Transfer and Thermal-Stress Analyses
Thermal-stress analysis of a disc brake 5.1.1
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CONTENTS
A sequentially coupled thermal-mechanical analysis of a disc brake with an Eulerian
approach 5.1.2
Exhaust manifold assemblage 5.1.3
Coolant manifold cover gasketed joint 5.1.4
Conductive, convective, and radiative heat transfer in an exhaust manifold 5.1.5 Thermal-stress analysis of a reactor pressure vessel bolted closure 5.1.6
6. Fluid Dynamics and Fluid-Structure Interaction
Conjugate heat transfer analysis of a component-mounted electronic circuit board 6.1.1
7. Electromagnetic Analyses Piezoelectric analyses
Eigenvalue analysis of a piezoelectric transducer 7.1.1 Transient dynamic nonlinear response of a piezoelectric transducer 7.1.2 Joule heating analyses
Thermal-electrical modeling of an automotive fuse 7.2.1
8. Mass Diffusion Analyses
Hydrogen diffusion in a vessel wall section 8.1.1
Diffusion toward an elastic crack tip 8.1.2
9. Acoustic and Shock Analyses
Fully and sequentially coupled acoustic-structural analysis of a muffler 9.1.1 Coupled acoustic-structural analysis of a speaker 9.1.2 Response of a submerged cylinder to an underwater explosion shock wave 9.1.3 Convergence studies for shock analyses using shell elements 9.1.4
UNDEX analysis of a detailed submarine model 9.1.5
Coupled acoustic-structural analysis of a pick-up truck 9.1.6 Long-duration response of a submerged cylinder to an underwater explosion 9.1.7 Deformation of a sandwich plate under CONWEP blast loading 9.1.8
10. Soils Analyses
Plane strain consolidation 10.1.1
Calculation of phreatic surface in an earth dam 10.1.2
Axisymmetric simulation of an oil well 10.1.3
Analysis of a pipeline buried in soil 10.1.4
Hydraulically induced fracture in a well bore 10.1.5
CONTENTS
11. Structural Optimization Analyses Topology optimization analyses
Topology optimization of an automotive control arm 11.1.1 Shape optimization analyses
Shape optimization of a connecting rod 11.2.1
12. Abaqus/Aqua Analyses
Jack-up foundation analyses 12.1.1
Riser dynamics 12.1.2
13. Design Sensitivity Analyses Overview
Design sensitivity analysis: overview 13.1.1
Examples
Design sensitivity analysis of a composite centrifuge 13.2.1 Design sensitivities for tire inflation, footprint, and natural frequency analysis 13.2.2 Design sensitivity analysis of a windshield wiper 13.2.3 Design sensitivity analysis of a rubber bushing 13.2.4
14. Postprocessing of Abaqus Results Files
User postprocessing of Abaqus results files: overview 14.1.1 Joining data from multiple results files and converting file format: FJOIN 14.1.2 Calculation of principal stresses and strains and their directions: FPRIN 14.1.3 Creation of a perturbed mesh from original coordinate data and eigenvectors: FPERT 14.1.4 Output radiation viewfactors and facet areas: FRAD 14.1.5 Creation of a data file to facilitate the postprocessing of elbow element results:
FELBOW 14.1.6
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INTRODUCTION
1.0 INTRODUCTION
This is the Example Problems Manual for Abaqus. It contains many solved examples that illustrate the use of the program for common types of problems. Some of the problems are quite difficult and require combinations of the capabilities in the code.
The problems have been chosen to serve two purposes: to verify the capabilities in Abaqus by exercising the code on nontrivial cases and to provide guidance to users who must work on a class of problems with which they are relatively unfamiliar. In each worked example the discussion in the manual states why the example is included and leads the reader through the standard approach to an analysis: element and mesh selection, material model, and a discussion of the results. Many of these problems are worked with different element types, mesh densities, and other variations.
Input data files for all of the analyses are included with the Abaqus release in compressed archive files. The abaqus fetch utility is used to extract these input files for use. For example, to fetch input file boltpipeflange_3d_cyclsym.inp, type
abaqus fetch job=boltpipeflange_3d_cyclsym.inp
Parametric study script (.psf) and user subroutine (.f) files can be fetched in the same manner. All files for a particular problem can be obtained by leaving off the file extension. The abaqus fetch utility is explained in detail in “Fetching sample input files,” Section 3.2.13 of the Abaqus Analysis User’s Manual.
It is sometimes useful to search the input files. The findkeyword utility is used to locate input files that contain user-specified input. This utility is defined in “Querying the keyword/problem database,” Section 3.2.12 of the Abaqus Analysis User’s Manual.
To reproduce the graphical representation of the solution reported in some of the examples, the output frequency used in the input files may need to be increased. For example, in “Linear analysis of the Indian Point reactor feedwater line,” Section 2.2.2, the figures that appear in the manual can be obtained only if the solution is written to the results file every increment; that is, if the input files are changed to read
*NODE FILE, ..., FREQUENCY=1 instead of FREQUENCY=100 as appears now.
In addition to the Example Problems Manual, there are two other manuals that contain worked problems. The Abaqus Benchmarks Manual contains benchmark problems (including the NAFEMS suite of test problems) and standard analyses used to evaluate the performance of Abaqus. The tests in this manual are multiple element tests of simple geometries or simplified versions of real problems. The Abaqus Verification Manual contains a large number of examples that are intended as elementary verification of the basic modeling capabilities.
The qualification process for new Abaqus releases includes running and verifying results for all problems in the Abaqus Example Problems Manual, the Abaqus Benchmarks Manual, and the Abaqus Verification Manual.
STATIC STRESS/DISPLACEMENT ANALYSES
1.
Static Stress/Displacement Analyses
•
“Static and quasi-static stress analyses,” Section 1.1•
“Buckling and collapse analyses,” Section 1.2•
“Forming analyses,” Section 1.3•
“Fracture and damage,” Section 1.4•
“Import analyses,” Section 1.5STATIC AND QUASI-STATIC STRESS ANALYSES
1.1
Static and quasi-static stress analyses
•
“Axisymmetric analysis of bolted pipe flange connections,” Section 1.1.1•
“Elastic-plastic collapse of a thin-walled elbow under in-plane bending and internal pressure,” Section 1.1.2•
“Parametric study of a linear elastic pipeline under in-plane bending,” Section 1.1.3•
“Indentation of an elastomeric foam specimen with a hemispherical punch,” Section 1.1.4•
“Collapse of a concrete slab,” Section 1.1.5•
“Jointed rock slope stability,” Section 1.1.6•
“Notched beam under cyclic loading,” Section 1.1.7•
“Uniaxial ratchetting under tension and compression,” Section 1.1.8•
“Hydrostatic fluid elements: modeling an airspring,” Section 1.1.9•
“Shell-to-solid submodeling and shell-to-solid coupling of a pipe joint,” Section 1.1.10•
“Stress-free element reactivation,” Section 1.1.11•
“Transient loading of a viscoelastic bushing,” Section 1.1.12•
“Indentation of a thick plate,” Section 1.1.13•
“Damage and failure of a laminated composite plate,” Section 1.1.14•
“Analysis of an automotive boot seal,” Section 1.1.15•
“Pressure penetration analysis of an air duct kiss seal,” Section 1.1.16•
“Self-contact in rubber/foam components: jounce bumper,” Section 1.1.17•
“Self-contact in rubber/foam components: rubber gasket,” Section 1.1.18•
“Submodeling of a stacked sheet metal assembly,” Section 1.1.19•
“Axisymmetric analysis of a threaded connection,” Section 1.1.20•
“Direct cyclic analysis of a cylinder head under cyclic thermal-mechanical loadings,” Section 1.1.21•
“Erosion of material (sand production) in an oil wellbore,” Section 1.1.22•
“Submodel stress analysis of pressure vessel closure hardware,” Section 1.1.23•
“Using a composite layup to model a yacht hull,” Section 1.1.24BOLTED PIPE JOINT
1.1.1 AXISYMMETRIC ANALYSIS OF BOLTED PIPE FLANGE CONNECTIONS
Product: Abaqus/Standard
A bolted pipe flange connection is a common and important part of many piping systems. Such connections are typically composed of hubs of pipes, pipe flanges with bolt holes, sets of bolts and nuts, and a gasket. These components interact with each other in the tightening process and when operation loads such as internal pressure and temperature are applied. Experimental and numerical studies on different types of interaction among these components are frequently reported. The studies include analysis of the bolt-up procedure that yields uniform bolt stress (Bibel and Ezell, 1992), contact analysis of screw threads (Fukuoka, 1992; Chaaban and Muzzo, 1991), and full stress analysis of the entire pipe joint assembly (Sawa et al., 1991). To establish an optimal design, a full stress analysis determines factors such as the contact stresses that govern the sealing performance, the relationship between bolt force and internal pressure, the effective gasket seating width, and the bending moment produced in the bolts. This example shows how to perform such a design analysis by using an economical axisymmetric model and how to assess the accuracy of the axisymmetric solution by comparing the results to those obtained from a simulation using a three-dimensional segment model. In addition, several three-dimensional models that use multiple levels of substructures are analyzed to demonstrate the use of substructures with a large number of retained degrees of freedom. Finally, a three-dimensional model containing stiffness matrices is analyzed to demonstrate the use of the matrix input functionality.
Geometry and model
The bolted joint assembly being analyzed is depicted in Figure 1.1.1–1. The geometry and dimensions of the various parts are taken from Sawa et al. (1991), modified slightly to simplify the modeling. The inner wall radius of both the hub and the gasket is 25 mm. The outer wall radii of the pipe flange and the gasket are 82.5 mm and 52.5 mm, respectively. The thickness of the gasket is 2.5 mm. The pipe flange has eight bolt holes that are equally spaced in the pitch circle of radius 65 mm. The radius of the bolt hole is modified in this analysis to be the same as that of the bolt: 8 mm. The bolt head (bearing surface) is assumed to be circular, and its radius is 12 mm.
The Young’s modulus is 206 GPa and the Poisson’s ratio is 0.3 for both the bolt and the pipe hub/flange. The gasket is modeled with either solid continuum or gasket elements. When continuum elements are used, the gasket’s Young’s modulus, E, equals 68.7 GPa and its Poisson’s ratio, , equals 0.3.
When gasket elements are used, a linear gasket pressure/closure relationship is used with the effective “normal stiffness,” , equal to the material Young’s modulus divided by the thickness so that 27.48 GPa/mm. Similarly a linear shear stress/shear motion relationship is used with an effective shear stiffness, , equal to the material shear modulus divided by the thickness so that 10.57 GPa/mm. The membrane behavior is specified with a Young’s modulus of 68.7 GPa and a Poisson’s ratio of 0.3. Sticking contact conditions are assumed in all contact areas: between the bearing surface and the flange and between the gasket and the hub. Contact between the bolt shank and the bolt hole is ignored.
BOLTED PIPE JOINT
The finite element idealizations of the symmetric half of the pipe joint are shown in Figure 1.1.1–2 and Figure 1.1.1–3, corresponding to the axisymmetric and three-dimensional analyses, respectively. The mesh used for the axisymmetric analysis consists of a mesh for the pipe hub/flange and gasket and a separate mesh for the bolts. In Figure 1.1.1–2 the top figure shows the mesh of the pipe hub and flange, with the bolt hole area shown in a lighter shade; and the bottom figure shows the overall mesh with the gasket and the bolt in place.
For the axisymmetric model second-order elements with reduced integration, CAX8R, are used throughout the mesh of the pipe hub/flange. The gasket is modeled with either CAX8R solid continuum elements or GKAX6 gasket elements. Contact between the gasket and the pipe hub/flange is modeled with contact pairs between surfaces defined on the faces of elements in the contact region or between such element-based surfaces and node-based surfaces. In an axisymmetric analysis the bolts and the perforated flange must be modeled properly. The bolts are modeled as plane stress elements since they do not carry hoop stress. Second-order plane stress elements with reduced integration, CPS8R, are employed for this purpose. The contact surface definitions, which are associated with the faces of the elements, account for the plane stress condition automatically. To account for all eight bolts used in the joint, the combined cross-sectional areas of the shank and the head of the bolts must be calculated and redistributed to the bolt mesh appropriately using the area attributes for the solid elements. The contact area is adjusted automatically.
Figure 1.1.1–4 illustrates the cross-sectional views of the bolt head and the shank. Each plane stress element represents a volume that extends out of the x–y plane. For example, element A represents a volume calculated as ( ) × ( ). Likewise, element B represents a volume calculated as ( ) × ( ). The sectional area in the x–z plane pertaining to a given element can be calculated as
where R is the bolt head radius, , or the shank radius, (depending on the element location), and and are x-coordinates of the left and right side of the given element, respectively.
If the sectional areas are divided by the respective element widths, and , we obtain representative element thicknesses. Multiplying each element thickness by eight (the number of bolts in the model) produces the thickness values that are found in the *SOLID SECTION options.
Sectional areas that are associated with bolt head elements located on the model’s contact surfaces are used to calculate the surface areas of the nodes used in defining the node-based surfaces of the model. Referring again to Figure 1.1.1–4, nodal contact areas for a single bolt are calculated as follows:
1.1.1–2
Abaqus Version 6.11 ID: Printed on:
BOLTED PIPE JOINT
where through are contact areas that are associated with contact nodes 1–9 and through are sectional areas that are associated with bolt head elements C–F. Multiplying the above areas by eight (the number of bolts in the model) provides the nodal contact areas found under the *SURFACE INTERACTION options.
A common way of handling the presence of the bolt holes in the pipe flange in axisymmetric analyses is to smear the material properties used in the bolt hole area of the mesh and to use inhomogeneous material properties that correspond to a weaker material in this region. General guidelines for determining the effective material properties for perforated flat plates are found in ASME Section VIII Div 2 Article 4–9. For the type of structure under study, which is not a flat plate, a common approach to determining the effective material properties is to calculate the elasticity moduli reduction factor, which is the ratio of the ligament area in the pitch circle to the annular area of the pitch circle. In this model the annular area of the pitch circle is given by 6534.51 mm2, and the total area of the bolt holes is given by 1608.5 mm2. Hence, the reduction factor is simply 0.754. The effective in-plane moduli of elasticity, and , are obtained by multiplying the respective moduli, and , by this factor. We assume material isotropy in the r–z plane; thus, The modulus in the hoop direction, , should be very small and is chosen such that 106. The in-plane shear modulus is then calculated based on the effective elasticity modulus: The shear moduli in the hoop direction are also calculated similarly but with set to zero (they are not used in an axisymmetric model). Hence, we have 155292 MPa, 0.155292 MPa, 59728 MPa, and 0.07765 MPa. These elasticity moduli are specified using *ELASTIC, TYPE=ENGINEERING CONSTANTS for the bolt hole part of the mesh.
The mesh for the three-dimensional analysis without substructures, shown in Figure 1.1.1–3, represents a 22.5° segment of the pipe joint and employs second-order brick elements with reduced integration, C3D20R, for the pipe hub/flange and bolts. The gasket is modeled with C3D20R elements or GK3D18 elements. The top figure shows the mesh of the pipe hub and flange, and the bottom figure shows both the gasket and bolt (in the lighter color). Contact is modeled by the interaction of contact surfaces defined by grouping specific faces of the elements in the contacting regions. For three-dimensional contact where both the master and slave surfaces are deformable, the SMALL SLIDING parameter must be used on the *CONTACT PAIR option to indicate that small relative sliding occurs between contacting surfaces. No special adjustments need be made for the material properties used in the three-dimensional model because all parts are modeled appropriately.
Four different meshes that use substructures to model the flange are tested. A first-level substructure is created for the entire 22.5° segment of the flange shown in Figure 1.1.1–3, while the gasket and the bolt are meshed as before. The nodes on the flange in contact with the bolt cap form a node-based surface, while the nodes on the flange in contact with the gasket form another node-based surface. These node-based surfaces will form contact pairs with the master surfaces on the bolt cap and on the gasket, which are defined with *SURFACE as before. The retained degrees of freedom on the substructure include all three degrees of freedom for the nodes in these node-based surfaces as well as for the nodes on the 0° and 22.5° faces of the flange. Appropriate boundary conditions are specified at the substructure usage level.
BOLTED PIPE JOINT
A second-level substructure of 45° is created by reflecting the first-level substructure with respect to the 22.5° plane. The nodes on the 22.5° face belonging to the reflected substructure are constrained in all three degrees of freedom to the corresponding nodes on the 22.5° face belonging to the original first-level substructure. The half-bolt and the gasket sector corresponding to the reflected substructure are also constructed by reflection. The retained degrees of freedom include all three degrees of freedom of all contact node sets and of the nodes on the 0° and 45° faces of the flange. MPC-type CYCLSYM is used to impose cyclic symmetric boundary conditions on these two faces.
A third-level substructure of 90° is created by reflecting the original 45° second-level substructure with respect to the 45° plane and by connecting it to the original 45° substructure. The remaining part of the gasket and the bolts corresponding to the 45°–90° sector of the model is created by reflection and appropriate constraints. In this case it is not necessary to retain any degrees of freedom on the 0° and 90° faces of the flange because this 90° substructure will not be connected to other substructures and appropriate boundary conditions can be specified at the substructure creation level.
The final substructure model is set up by mirroring the 90° mesh with respect to the symmetry plane of the gasket perpendicular to the y-axis. Thus, an otherwise large analysis ( 750,000 unknowns) when no substructures are used can be solved conveniently ( 80,000 unknowns) by using the third-level substructure twice. The sparse solver is used because it significantly reduces the run time for this model. Finally, a three-dimensional matrix-based model is created by replacing elements for the entire 22.5° segment of the flange shown in Figure 1.1.1–3 with stiffness matrices, while the gasket and the bolt are meshed as before. Contact between the flange and gasket and the flange and bolt cap is modeled using node-based slave surfaces just as for the substructure models. Appropriate boundary conditions are applied as in the three-dimensional model without substructures.
Loading and boundary conditions
The only boundary conditions are symmetry boundary conditions. In the axisymmetric model 0 is applied to the symmetry plane of the gasket and to the bottom of the bolts. In the three-dimensional model 0 is applied to the symmetry plane of the gasket as well as to the bottom of the bolt. The 0° and 22.5° planes are also symmetry planes. On the 22.5° plane, symmetry boundary conditions are enforced by invoking suitable nodal transformations and applying boundary conditions to local directions in this symmetry plane. These transformations are implemented using the *TRANSFORM option. On both the symmetry planes, the symmetry boundary conditions 0 are imposed everywhere except for the dependent nodes associated with the C BIQUAD MPC and nodes on one side of the contact surface. The second exception is made to avoid overconstraining problems, which arise if there is a boundary condition in the same direction as a Lagrange multiplier constraint associated with the *FRICTION, ROUGH option.
In the models where substructures are used, the boundary conditions are specified depending on what substructure is used. For the first-level 22.5° substructure the boundary conditions and constraint equations are the same as for the three-dimensional model shown in Figure 1.1.1–3. For the 45° second-level substructure the symmetry boundary conditions are enforced on the 45° plane with the constraint equation 0. A transform could have been used as well. For the 90° third-level substructure the face 90° is constrained with the boundary condition 0.
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For the three-dimensional model containing matrices, nodal transformations are applied for symmetric boundary conditions. Entries in the stiffness matrices for these nodes are also in local coordinates.
A clamping force of 15 kN is applied to each bolt by using the *PRE-TENSION SECTION option. The pre-tension section is identified by means of the *SURFACE option. The pre-tension is then prescribed by applying a concentrated load to the pre-tension node. In the axisymmetric analysis the actual load applied is 120 kN since there are eight bolts. In the three-dimensional model with no substructures the actual load applied is 7.5 kN since only half of a bolt is modeled. In the models using substructures all half-bolts are loaded with a 7.5 kN force. For all of the models the pre-tension section is specified about halfway down the bolt shank.
Sticking contact conditions are assumed in all surface interactions in all analyses and are simulated with the *FRICTION, ROUGH and *SURFACE BEHAVIOR, NO SEPARATION options.
Results and discussion
All analyses are performed as small-displacement analyses.
Figure 1.1.1–5 shows a top view of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when solid continuum elements are used to model the gasket. The figure shows that the compressive normal stress is highest at the outer edge of the gasket, decreases radially inward, and changes from compression to tension at a radius of about 35 mm, which is consistent with findings reported by Sawa et al. (1991). The close agreement in the overall solution between axisymmetric and three-dimensional analyses is quite apparent, indicating that, for such problems, axisymmetric analysis offers a simple yet reasonably accurate alternative to three-dimensional analysis. Figure 1.1.1–6 shows a top view of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when gasket elements are used to model the gasket. Close agreement in the overall solution between the axisymmetric and three-dimensional analyses is also seen in this case. The gasket starts carrying compressive load at a radius of about 40 mm, a difference of 5 mm with the previous result. This difference is the result of the gasket elements being unable to carry tensile loads in their thickness direction. This solution is physically more realistic since, in most cases, gaskets separate from their neighboring parts when subjected to tensile loading. Removing the *SURFACE BEHAVIOR, NO SEPARATION option from the gasket/flange contact surface definition in the input files that model the gasket with continuum elements yields good agreement with the results obtained in Figure 1.1.1–6 (since, in that case, the solid continuum elements in the gasket cannot carry tensile loading in the gasket thickness direction).
The models in this example can be modified to study other factors, such as the effective seating width of the gasket or the sealing performance of the gasket under operating loads. The gasket elements offer the advantage of allowing very complex behavior to be defined in the gasket thickness direction. Gasket elements can also use any of the small-strain material models provided in Abaqus including user-defined material models. Figure 1.1.1–7 shows a comparison of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when isotropic material properties are prescribed for gasket elements.
BOLTED PIPE JOINT
The results in Figure 1.1.1–7 compare well with the results in Figure 1.1.1–5 from analyses in which solid and axisymmetric elements are used to simulate the gasket.
Figure 1.1.1–8 shows the distribution of the normal stresses in the gasket at the interface in the plane 0. The results are plotted for the three-dimensional model containing only solid continuum elements and no substructures, for the three-dimensional model with matrices, and for the four models containing the substructures described above.
An execution procedure is available to combine model and results data from two substructure output databases into a single output database. For more information, see “Combining output from substructures,” Section 3.2.18 of the Abaqus Analysis User’s Manual.
This example can also be used to demonstrate the effectiveness of the quasi-Newton nonlinear solver. This solver utilizes an inexpensive, approximate stiffness matrix update for several consecutive equilibrium iterations, rather than a complete stiffness matrix factorization each iteration as used in the default full Newton method. The quasi-Newton method results in an increased number of less expensive iterations, and a net savings in computing cost.
Input files
boltpipeflange_axi_solidgask.inp Axisymmetric analysis containing a gasket modeled with solid continuum elements.
boltpipeflange_axi_node.inp Node definitions for boltpipeflange_axi_solidgask.inp and boltpipeflange_axi_gkax6.inp.
boltpipeflange_axi_element.inp Element definitions for
boltpipeflange_axi_solidgask.inp.
boltpipeflange_3d_solidgask.inp Three-dimensional analysis containing a gasket modeled with solid continuum elements.
boltpipeflange_axi_gkax6.inp Axisymmetric analysis containing a gasket modeled with gasket elements.
boltpipeflange_3d_gk3d18.inp Three-dimensional analysis containing a gasket modeled with gasket elements.
boltpipeflange_3d_substr1.inp Three-dimensional analysis using the first-level substructure (22.5° model).
boltpipeflange_3d_substr2.inp Three-dimensional analysis using the second-level substructure (45° model).
boltpipeflange_3d_substr3_1.inp Three-dimensional analysis using the third-level substructure once (90° model).
boltpipeflange_3d_substr3_2.inp Three-dimensional analysis using the third-level substructure twice (90° mirrored model).
boltpipeflange_3d_gen1.inp First-level substructure generation data referenced by boltpipeflange_3d_substr1.inp and
boltpipeflange_3d_gen2.inp.
boltpipeflange_3d_gen2.inp Second-level substructure generation data referenced by boltpipeflange_3d_substr2.inp and
boltpipeflange_3d_gen3.inp.
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boltpipeflange_3d_gen3.inp Third-level substructure generation data referenced by boltpipeflange_3d_substr3_1.inp and
boltpipeflange_3d_substr3_2.inp. boltpipeflange_3d_node.inp Nodal coordinates used in
boltpipeflange_3d_substr1.inp, boltpipeflange_3d_substr2.inp, boltpipeflange_3d_substr3_1.inp, boltpipeflange_3d_substr3_2.inp, boltpipeflange_3d_cyclsym.inp, boltpipeflange_3d_gen1.inp, boltpipeflange_3d_gen2.inp, and boltpipeflange_3d_gen3.inp.
boltpipeflange_3d_cyclsym.inp Same as file boltpipeflange_3d_substr2.inp except that CYCLSYM type MPCs are used.
boltpipeflange_3d_missnode.inp Same as file boltpipeflange_3d_gk3d18.inp except that the option to generate missing nodes is used for gasket elements.
boltpipeflange_3d_isomat.inp Same as file boltpipeflange_3d_gk3d18.inp except that gasket elements are modeled as isotropic using the *MATERIAL option.
boltpipeflange_3d_ortho.inp Same as file boltpipeflange_3d_gk3d18.inp except that gasket elements are modeled as orthotropic and the *ORIENTATION option is used.
boltpipeflange_axi_isomat.inp Same as file boltpipeflange_axi_gkax6.inp except that gasket elements are modeled as isotropic using the *MATERIAL option.
boltpipeflange_3d_usr_umat.inp Same as file boltpipeflange_3d_gk3d18.inp except that gasket elements are modeled as isotropic with user subroutine UMAT.
boltpipeflange_3d_usr_umat.f User subroutine UMAT used in boltpipeflange_3d_usr_umat.inp.
boltpipeflange_3d_solidnum.inp Same as file boltpipeflange_3d_gk3d18.inp except that solid element numbering is used for gasket elements. boltpipeflange_3d_matrix.inp Three-dimensional analysis containing matrices and a
gasket modeled with solid continuum elements.
boltpipeflange_3d_stiffPID4.inp Matrix representing stiffness of a part of the flange segment for three-dimensional analysis containing matrices.
boltpipeflange_3d_stiffPID5.inp Matrix representing stiffness of the remaining part of the flange segment for three-dimensional analysis containing matrices.
BOLTED PIPE JOINT
boltpipeflange_3d_qn.inp Same as file boltpipeflange_3d_gk3d18.inp except that the quasi-Newton nonlinear solver is used.
References
•
Bibel, G. D., and R. M. Ezell, “An Improved Flange Bolt-Up Procedure Using Experimentally Determined Elastic Interaction Coefficients,” Journal of Pressure Vessel Technology, vol. 114, pp. 439–443, 1992.•
Chaaban, A., and U. Muzzo, “Finite Element Analysis of Residual Stresses in Threaded End Closures,” Transactions of ASME, vol. 113, pp. 398–401, 1991.•
Fukuoka, T., “Finite Element Simulation of Tightening Process of Bolted Joint with a Tensioner,” Journal of Pressure Vessel Technology, vol. 114, pp. 433–438, 1992.•
Sawa, T., N. Higurashi, and H. Akagawa, “A Stress Analysis of Pipe Flange Connections,” Journal of Pressure Vessel Technology, vol. 113, pp. 497–503, 1991.1.1.1–8
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BOLTED PIPE JOINT 80 θ = π 4 d = 50 d = 105 d = 130 d = 165 r = 8 15 d = 105 d = 50 centerline Top View Side View 2.5 Gasket Bolt 24 16 10 26 47 20
BOLTED PIPE JOINT 1 2 3 1 2 3 1 2 3 1 2 3
Figure 1.1.1–2 Axisymmetric model of the bolted joint.
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BOLTED PIPE JOINT y 1 2 3 4 5 6 7 8 9 C D E F z x WB WA HA HB contact nodes area A area B element A element B TOP VIEW FRONT VIEW Rbolthead Rshank
Figure 1.1.1–4 Cross-sectional views of the bolt head and the shank.
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BOLTED PIPE JOINT 1 2 3 12 1 2 3 11 1 2 3 7 8 9 12 11 10 8 11 9 9 9 11 8 9 9 8 12 8 10 11 7 12 1 2 3 4 5 6 1 2 3 4 5 6 8 7 7 1 2 3 4 5 6 7 12 12 1 2 3 4 5 6 11 10 10 11 10 10 7 12 2 3 4 5 6 2 3 4 5 6 1 2 3 S22 VALUE 1 -1.00E+02 2 -8.90E+01 3 -7.81E+01 4 -6.72E+01 5 -5.63E+01 6 -4.54E+01 7 -3.45E+01 8 -2.36E+01 9 -1.27E+01 10 -1.81E+00 11 +9.09E+00 12 +2.00E+01 1 2 3 12 11 10 10 10 10 10 10 11 10 9 9 9 9 11 9 8 8 8 8 8 11 8 8 8 11 11 11 9 2 3 4 5 6 12 12 7 7 7 7 2 3 4 5 6 7 7 7 12 2 3 4 5 6 12 2 3 4 5 6 12 9 7 10 12 9 12 11 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 1 2 3 S22 VALUE 1 -1.00E+02 2 -8.90E+01 3 -7.81E+01 4 -6.72E+01 5 -5.63E+01 6 -4.54E+01 7 -3.45E+01 8 -2.36E+01 9 -1.27E+01 10 -1.81E+00 11 +9.09E+00 12 +2.00E+01
Figure 1.1.1–5 Normal stress distribution in the gasket contact surface when solid elements are used to model the gasket: three-dimensional versus axisymmetric results.
BOLTED PIPE JOINT 1 2 3 3 7 8 9 1011 5 6 4 4 4 4 4 4 3 7 8 9 1011 7 8 9 1011 7 8 9 1011 5 6 5 6 5 6 4 5 6 3 5 6 5 6 3 3 3 3 3 7 8 9 1011 4 7 8 9 1011 7 8 91011 78 91011 5 6 4 4 4 4 4 4 4 5 6 5 6 5 6 5 6 5 6 5 6 5 6 3 4 5 6 7 8 9 1011 7 8 9 1011 7 8 9 1011 7 8 9 1011 7 8 9 1011 7 8 9 1011 7 8 91011 4 78 91011 3 3 3 3 3 3 3 1 2 3 S11 VALUE 1 -2.00E+01 2 -9.09E+00 3 +1.82E+00 4 +1.27E+01 5 +2.36E+01 6 +3.45E+01 7 +4.55E+01 8 +5.64E+01 9 +6.73E+01 10 +7.82E+01 11 +8.91E+01 12 +1.00E+02 1 2 3 7 8 9 3 7 8910 11 7 89 11 8 91011 4 3 3 3 4 6 5 5 6 6 8 7 8 10 7 8910 7 11 7 8 11 7 8 10 5 6 4 3 5 6 3 3 4 5 6 4 5 6 5 6 5 6 3 78 91011 78 910 78 91011 3 78 910 11 78 910 78 91011 3 4 4 4 1 2 3 S11 VALUE 1 -2.00E+01 2 -9.09E+00 3 +1.82E+00 4 +1.27E+01 5 +2.36E+01 6 +3.45E+01 7 +4.55E+01 8 +5.64E+01 9 +6.73E+01 10 +7.82E+01 11 +8.91E+01 12 +1.00E+02
Figure 1.1.1–6 Normal stress distribution in the gasket contact surface when gasket elements are used with direct specification of the gasket behavior: three-dimensional versus axisymmetric results.
1.1.1–14
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BOLTED PIPE JOINT 1 2 3 2 3 4 5 6 9 9 77 10 10 10 2 3 4 5 6 2 3 4 5 6 8 8 8 2 3 4 5 6 7 2 3 4 5 6 7 7 8 8 65432 2 3 4 5 6 9 7 12 7 12 1111 8 2 3 4 5 6 11 6 5432 8 11 10 10 2 12 12 9 9 8 8 10 10 11 11 7 7 9 9 10 10 12 12 9 12 9 11 11 12 12 11 12 10 9 7 6 543 11 8 7 7 66 554433 12 12 8 11 10 10 9 11 9 8 2 3 4 5 6 8 9 7 10 2 3 4 5 6 8 10 9 7 8 32 4 5 6 7 9 10 12 2 3 4 5 6 11 7 11 12 8 9 10 12 2 3 4 5 6 10 7 12 11 11 8 9 11 12 3 4 5 6 7 8 9 10 11 12 1 2 3 S11 VALUE 1 -1.00E+02 2 -8.91E+01 3 -7.82E+01 4 -6.73E+01 5 -5.64E+01 6 -4.55E+01 7 -3.45E+01 8 -2.36E+01 9 -1.27E+01 10 -1.82E+00 11 +9.09E+00 12 +2.00E+01 1 2 3 2 3 4 5 8 9 6 7 10 2 3 4 5 8 10 9 6 7 8 432 5 6 7 2 3 4 5 12 9 11 2 3 4 5 10 12 8 6 7 8 11 9 10 12 11 11 6 7 10 8 12 11 9 9 432 5 12 10 6 7 11 11 10 9 8 432 5 6 7 12 11 12 10 9 12 8 6 7 2 3 4 5 2 3 4 5 10 9 8 6 7 11 12 2 3 4 5 12 11 10 9 8 7 6 2 3 4 5 8 6 7 9 10 2 3 4 5 8 10 9 6 7 8 432 5 6 7 2 3 4 5 12 9 11 10 2 3 4 5 12 6 7 8 8 11 9 12 10 11 9 6 7 10 8 12 11 11 9 432 5 12 10 6 7 2 3 4 5 11 10 9 8 6 7 2 3 4 5 12 11 12 10 9 8 12 6 7 11 1 2 3 S11 VALUE 1 -1.00E+02 2 -8.91E+01 3 -7.82E+01 4 -6.73E+01 5 -5.64E+01 6 -4.55E+01 7 -3.45E+01 8 -2.36E+01 9 -1.27E+01 10 -1.82E+00 11 +9.09E+00 12 +2.00E+01
Figure 1.1.1–7 Normal stress distribution in the gasket contact surface when gasket elements are used with isotropic material properties: three-dimensional versus axisymmetric results.
BOLTED PIPE JOINT 22.5_matrix 22.5_no_sup 22.5_sup 45_sup 90_sup 90r_sup
Figure 1.1.1–8 Normal stress distribution in the gasket contact surface along the line 0 for the models with and without substructures.
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1.1.2 ELASTIC-PLASTIC COLLAPSE OF A THIN-WALLED ELBOW UNDER IN-PLANE BENDING AND INTERNAL PRESSURE
Product: Abaqus/Standard
Elbows are used in piping systems because they ovalize more readily than straight pipes and, thus, provide flexibility in response to thermal expansion and other loadings that impose significant displacements on the system. Ovalization is the bending of the pipe wall into an oval—i.e., noncircular—configuration. The elbow is, thus, behaving as a shell rather than as a beam. Straight pipe runs do not ovalize easily, so they behave essentially as beams. Thus, even under pure bending, complex interaction occurs between an elbow and the adjacent straight pipe segments; the elbow causes some ovalization in the straight pipe runs, which in turn tend to stiffen the elbow. This interaction can create significant axial gradients of bending strain in the elbow, especially in cases where the elbow is very flexible. This example provides verification of shell and elbow element modeling of such effects, through an analysis of a test elbow for which experimental results have been reported by Sobel and Newman (1979). An analysis is also included with elements of type ELBOW31B (which includes ovalization but neglects axial gradients of strain) for the elbow itself and beam elements for the straight pipe segments. This provides a comparative solution in which the interaction between the elbow and the adjacent straight pipes is neglected. The analyses predict the response up to quite large rotations across the elbow, so as to investigate possible collapse of the pipe and, particularly, the effect of internal pressure on that collapse.
Geometry and model
The elbow configuration used in the study is shown in Figure 1.1.2–1. It is a thin-walled elbow with elbow factor
and radius ratio 3.07, so the flexibility factor from Dodge and Moore (1972) is 10.3. (The flexibility factor for an elbow is the ratio of the bending flexibility of an elbow segment to that of a straight pipe of the same dimensions, for small displacements and elastic response.) This is an extremely flexible case because the pipe wall is so thin.
To demonstrate convergence of the overall moment-rotation behavior with respect to meshing, the two shell element meshes shown in Figure 1.1.2–2 are analyzed. Since the loading concerns in-plane bending only, it is assumed that the response is symmetric about the midplane of the system so that in the shell element model only one-half of the system need be modeled. Element type S8R5 is used, since tests have shown this to be the most cost-effective shell element in Abaqus (input files using element types S9R5, STRI65, and S8R for this example are included with the Abaqus release). The elbow element meshes replace each axial division in the coarser shell element model with one ELBOW32 or two ELBOW31 elements and use 4 or 6 Fourier modes to model the deformation around the pipe. Seven integration points are used through the pipe wall in all the analyses. This is usually adequate to
ELASTIC-PLASTIC COLLAPSE
provide accurate modeling of the progress of yielding through the section in such cases as these, where essentially monotonic straining is expected.
The ends of the system are rigidly attached to stiff plates in the experiments. These boundary conditions are easily modeled for the ELBOW elements and for the fixed end in the shell element model. For the rotating end of the shell element model the shell nodes must be constrained to a beam node that represents the motion of the end plate. This is done using the *KINEMATIC COUPLING option as described below.
The material is assumed to be isotropic and elastic-plastic, following the measured response of type 304 stainless steel at room temperature, as reported by Sobel and Newman (1979). Since all the analyses give results that are stiffer than the experimentally measured response, and the mesh convergence tests (results are discussed below) demonstrate that the meshes are convergent with respect to the overall response of the system, it seems that this stress-strain model may overestimate the material’s actual strength.
Loading
The load on the pipe has two components: a “dead” load, consisting of internal pressure (with a closed end condition), and a “live” in-plane bending moment applied to the end of the system. The pressure is applied to the model in an initial step and then held constant in the second analysis step while the bending moment is increased. The pressure values range from 0.0 to 3.45 MPa (500 lb/in2), which is the range of interest for design purposes. The equivalent end force associated with the closed-end condition is applied as a follower force because it rotates with the motion of the end plane.
Kinematic boundary conditions
The fixed end of the system is assumed to be fully built-in. The loaded end is fixed into a very stiff plate. For the ELBOW element models this condition is represented by the NODEFORM boundary condition applied at this node. In the shell element model this rigid plate is represented by a single node, and the shell nodes at the end of the pipe are attached to it by using a kinematic coupling constraint and specifying that all degrees of freedom at the shell nodes are constrained to the motion of the single node.
Results and discussion
The moment-rotation responses predicted by the various analysis models and measured in the experiment, all taken at zero internal pressure, are compared in Figure 1.1.2–3. The figure shows that the two shell models give very similar results, overestimating the experimentally measured collapse moment by about 15%. The 6-mode ELBOW element models are somewhat stiffer than the shell models, and those with 4 Fourier modes are much too stiff. This clearly shows that, for this very flexible system, the ovalization of the elbow is too localized for even the 6-mode ELBOW representation to provide accurate results.
Since we know that the shell models are convergent with respect to discretization, the most likely explanation for the excessive stiffness in comparison to the experimentally measured response is that the material model used in the analyses is too strong. Sobel and Newman (1979) point out that the stress-strain curve measured and used in this analysis, shown in Figure 1.1.2–1, has a 0.2% offset yield that is 20% higher than the Nuclear Systems Materials Handbook value for type 304 stainless
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steel at room temperature, which suggests the possibility that the billets used for the stress-strain curve measurement may have been taken from stronger parts of the fabrication. If this is the case, it points out the likelihood that the elbow tested is rather nonuniform in strength properties in spite of the care taken in its manufacture. We are left with the conclusion that discrepancies of this magnitude cannot be eliminated in practical cases, and the design use of such analysis results must allow for them.
Figure 1.1.2–4 compares the moment-rotation response for opening and closing moments under 0 and 3.45 MPa (500 lb/in2) internal pressure and shows the strong influence of large-displacement effects. If large-displacement effects were not important, the opening and closing moments would produce the same response. However, even with a 1° relative rotation across the elbow assembly, the opening and closing moments differ by about 12%; with a 2° relative rotation, the difference is about 17%. Such magnitudes of relative rotation would not normally be considered large; in this case it is the coupling into ovalization that makes geometric nonlinearity significant. As the rotation increases, the cases with closing moment loading show collapse, while the opening moment curves do not. In both cases internal pressure shows a strong effect on the results, which is to be expected in such a thin-walled pipeline. The level of interaction between the straight pipe and the elbows is well illustrated by the strain distribution on the outside wall, shown in Figure 1.1.2–5. The strain contours are slightly discontinuous at the ends of the curved elbow section because the shell thickness changes at those sections.
Figure 1.1.2–6 shows a summary of the results from this example and “Uniform collapse of straight and curved pipe segments,” Section 1.1.5 of the Abaqus Benchmarks Manual. The plot shows the collapse value of the closing moment under in-plane bending as a function of internal pressure. The strong influence of pressure on collapse is apparent. In addition, the effect of analyzing the elbow by neglecting interaction between the straight and curved segments is shown: the “uniform bending” results are obtained by using elements of type ELBOW31B in the bend and beams (element type B31) for the straight segments. The importance of the straight/elbow interaction is apparent. In this case the simpler analysis neglecting the interaction is conservative (in that it gives consistently lower values for the collapse moment), but this conservatism cannot be taken for granted. The analysis of Sobel and Newman (1979) also neglects interaction and agrees quite well with the results obtained here.
For comparison the small-displacement limit analysis results of Goodall (1978), as well as his large-displacement, elastic-plastic lower bound (Goodall, 1978a), are also shown in this figure. Again, the importance of large-displacement effects is apparent from that comparison.
Detailed results obtained with the model that uses ELBOW31 elements are shown in Figure 1.1.2–7 through Figure 1.1.2–9. Figure 1.1.2–7 shows the variation of the Mises stress along the length of the piping system. The length is measured along the centerline of the pipe starting at the loaded end. The figure compares the stress distribution at the intrados (integration point 1) on the inner and outer surfaces of the elements (section points 1 and 7, respectively). Figure 1.1.2–8 shows the variation of the Mises stress around the circumference of two elements (451 and 751) that are located in the bend section of the model; the results are for the inner surface of the elements (section point 1). Figure 1.1.2–9 shows the ovalization of elements 451 and 751. A nonovalized, circular cross-section is included in the figure for comparison. From the figure it is seen that element 751, located at the center of the bend section, experiences the most severe ovalization. These three figures were produced with the aid of the elbow element postprocessing program felbow.f (“Creation of a data file to facilitate the postprocessing of elbow element results: FELBOW,” Section 14.1.6), written in FORTRAN. The postprocessing programs
ELASTIC-PLASTIC COLLAPSE
felbow.C(“A C++ version of FELBOW,” Section 10.15.6 of the Abaqus Scripting User’s Manual) and felbow.py (“An Abaqus Scripting Interface version of FELBOW,” Section 9.10.12 of the Abaqus Scripting User’s Manual), written in C++ and Python, respectively, are also available for generating the data for figures such as Figure 1.1.2–8 and Figure 1.1.2–9. The user must ensure that the output variables are written to the output database to use these two programs.
Shell-to-solid submodeling
One particular case is analyzed using the shell-to-solid submodeling technique. This problem verifies the interpolation scheme in the case of double curved surfaces. A solid submodel using C3D27R elements is created around the elbow part of the pipe, spanning an angle of 40°. The finer submodel mesh has three elements through the thickness, 10 elements around half of the circumference of the cylinder, and 10 elements along the length of the elbow. Both ends are driven from the global shell model made of S8R elements. The time scale of the static submodel analysis corresponds to the arc length in the global Riks analysis. The submodel results agree closely with the shell model. The *SECTION FILE option is used to output the total force and the total moment in a cross-section through the submodel.
Shell-to-solid coupling
A model using the shell-to-solid coupling capability in Abaqus is included. Such a model can be used for a careful study of the stress and strain fields in the elbow. The entire elbow is meshed with C3D20R elements, and the straight pipe sections are meshed with S8R elements (see Figure 1.1.2–10). At each shell-to-solid interface illustrated in Figure 1.1.2–10, an element-based surface is defined on the edge of the solid mesh and an edge-based surface is defined on the edge of the shell mesh. The *SHELL TO SOLID COUPLING option is used in conjunction with these surfaces to couple the shell and solid meshes.
Edge-based surfaces are defined at the end of each pipe segment. These surfaces are coupled to reference nodes that are defined at the center of the pipes using the *COUPLING option in conjunction with the *DISTRIBUTING option. The loading and fixed boundary conditions are applied to the reference points. The advantage of using this method is that the pipe cross-sectional areas are free to deform; thus, ovalization at the ends is not constrained. The moment-rotation response of the shell-to-solid coupling model agrees very well with the results shown in Figure 1.1.2–4.
Input files
In all the following input files (with the exception of elbowcollapse_elbow31b_b31.inp, elbowcollapse_s8r5_fine.inp, and elbowcolpse_shl2sld_s8r_c3d20r.inp) the step concerning the application of the pressure load is commented out. To include the effects of the internal pressure in any given analysis, uncomment the step definition in the appropriate input file.
elbowcollapse_elbow31b_b31.inp ELBOW31B and B31 element model. elbowcollapse_elbow31_6four.inp ELBOW31 model with 6 Fourier modes. elbowcollapse_elbow32_6four.inp ELBOW32 model with 6 Fourier modes. elbowcollapse_s8r.inp S8R element model.
1.1.2–4
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elbowcollapse_s8r5.inp S8R5 element model. elbowcollapse_s8r5_fine.inp Finer S8R5 element model. elbowcollapse_s9r5.inp S9R5 element model. elbowcollapse_stri65.inp STRI65 element model.
elbowcollapse_submod.inp Submodel using C3D27R elements.
elbowcolpse_shl2sld_s8r_c3d20r.inp Shell-to-solid coupling model using S8R and C3D20R elements.
References
•
Dodge, W. G., and S. E. Moore, “Stress Indices and Flexibility Factors for Moment Loadings on Elbows and Curved Pipes,” Welding Research Council Bulletin, no. 179, 1972.•
Goodall, I. W., “Lower Bound Limit Analysis of Curved Tubes Loaded by Combined Internal Pressure and In-Plane Bending Moment,” Research Division Report RD/B/N4360, Central Electricity Generating Board, England, 1978.•
Goodall, I. W., “Large Deformations in Plastically Deforming Curved Tubes Subjected to In-Plane Bending,” Research Division Report RD/B/N4312, Central Electricity Generating Board, England, 1978a.•
Sobel, L. H., and S. Z. Newman, “Elastic-Plastic In-Plane Bending and Buckling of an Elbow: Comparison of Experimental and Simplified Analysis Results,” Westinghouse Advanced Reactors Division, Report WARD–HT–94000–2, 1979.ELASTIC-PLASTIC COLLAPSE 400 300 200 100 0 0 1 2 3 4 5 0 10 20 30 40 50 60 70 Strain, % Stress, MPa Stress, 10 3 lb/in 2 1.83 m (72.0 in)
Moment applied here 610 mm (24.0 in) 407 mm (16.02 in) 10.4 mm (0.41 in) thickness Young's modulus: Poisson's ratio: 193 GPa (28 x 106 lb/in2 ) 0.2642
Figure 1.1.2–1 MLTF elbow: geometry and measured material response.
1.1.2–6
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ELASTIC-PLASTIC COLLAPSE 0.04 0.08 0.12 0.16 0.20 0.24 0 2 0.04 0.08 0.12 0.16 0.20 0.24 0 3 0.04 0.08 0.12 0.16 0.20 0.24 0 4 0.04 0.08 0.12 0.16 0.20 0.24 0 5,7 0.04 0.08 0.12 0.16 0.20 0.24 0 6 0.04 0.08 0.12 0.16 0.20 0.24 0 8 0.04 0.08 0.12 0.16 0.20 0.24 0 10 0.04 0.08 0.12 0.16 0.20 0.24 0 0.04 0.08 0.12 0.16 0.20 0.24 0 0.04 0.08 0.12 0.16 0.20 0.24 0 1.0 2.0 1.0 4.0 7.0 10.0 13.0 11 0 1 Line variable 1 Experiment 2 S8R5 3 S8R5-finer mesh 4 ELBOW32 - 6 mode 5 ELBOW32 - 4 mode 6 ELBOW31 - 6 mode 7 ELBOW31 - 4 mode 8 ELBOW31 - Coarse 6 9 ELBOW31 - Coarse 4 10 ELBOW31B - 6 mode 11 ELBOW31B - 4 mode
End rotation, rad End rotation, deg
9 Moment, kN-m 200 150 100 50 Moment, 10 6 lb-in
Figure 1.1.2–3 Moment-rotation response: mesh convergence studies.
0.00 0.04 0.08 0.12 0.16 0.20 0.24 [x10 ]6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
End Rotation, rad
Moment, lb-in ELBOW31closing/0 ELBOW31closing/500 ELBOW31opening/0 ELBOW31opening/500 S8R5closing/0 S8R5closing/500 S8R5opening/0 S8R5opening/500
Figure 1.1.2–4 Moment-rotation response: pressure dependence.
1.1.2–8
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ELASTIC-PLASTIC COLLAPSE E22 VALUE -1.56E-02 -1.35E-02 -1.14E-02 -9.40E-03 -7.33E-03 -5.26E-03 -3.19E-03 -1.12E-03 +9.47E-04 +3.01E-03 +5.08E-03 +7.15E-03 +9.22E-03
+1.12E-02 Hoop strain
E11 VALUE -1.36E-02 -1.02E-02 -6.82E-03 -3.43E-03 -4.61E-05 +3.34E-03 +6.73E-03 +1.01E-02 +1.35E-02 +1.69E-02 +2.02E-02 +2.36E-02 +2.70E-02 +3.04E-02 Axial strain
ELASTIC-PLASTIC COLLAPSE 250 500 750 1000 1.2 1.4 1.6 1.8 2.0 2.2 2.4 1.0 0 1 2 3 4 5
Internal pressure, MPa 275 250 225 200 175 150 125 S8R5 ELBOW31 ELBOW31B Goodall (1978a), large displacement
elastic-plastic lower bound
Sobel and Newman (1979), uniform bending analysis
Internal pressure, lb/in2
Collapse moment, kN-m
Collapse moment, 10
6 lb-in
Goodall(1978), small– displacement limit analysis
6
Figure 1.1.2–6 In-plane bending of an elbow, elastic-plastic collapse moment results.
0. 50. 100.
Length along pipe, in 10.
20. 30. 40. 50.
Mises stress, psi
[x103] XMIN 1.500E+00 XMAX 1.322E+02 YMIN 4.451E+03 YMAX 5.123E+04 MISES_I MISES_O
Figure 1.1.2–7 Mises stress distribution along the length of the piping system.
1.1.2–10
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0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. Length around element circumference, in
25. 30. 35. 40. 45. 50. 55. 60.
Mises stress, psi
[x103] XMIN 0.000E+00 XMAX 4.892E+01 YMIN 2.635E+04 YMAX 5.778E+04 MISES451 MISES751
Figure 1.1.2–8 Mises stress distribution around the circumference of elements 451 and 751.
-10. -5. 0. 5. 10. Local x-axis -10. -5. 0. 5. 10. Local y-axis XMIN -7.805E+00 XMAX 7.805E+00 YMIN -8.732E+00 YMAX 8.733E+00 CIRCLE OVAL_451 OVAL_751
ELASTIC-PLASTIC COLLAPSE 1 2 3 shell elements solid elements shell-to-solid interface
Figure 1.1.2–10 Shell-to-solid coupling model study.
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