application of boundary conditions to obtain better fea results

Application of Boundary Conditions

to Obtain Better FEA Results

Kee H. Lee, P.E. ([email protected])

Design & Structural QC Group

Civil Design Team

November 20, 2015

mailto:[email protected]

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Contents

I. Finite Element Method

II. Pre-requisition for Structural Analysis

III. Typical Boundary Conditions (B.C.)

IV. Element Mesh Generation

V. FE Analysis Boundary Based on Structural Behavior

VI. Examples of B.C. Applications

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Contents - cont.

Application of Boundary

Conditions

1. Finite Element

Method

Purpose

Fundamental Concepts

Discretization

Pre/Post-Processing

Advantages & Disadvantages

2. Pre-requisition for

Structural Analysis

3. Typical Boundary

Conditions (B.C.)

4. Element Mesh

Generation

5. FE Analysis

Boundary Based on

Structural Behavior

6. Examples of B.C.

Applications

Types of Structural Analysis

Element Types

Degree of Freedom

Element Coordinate Systems & Output Data

Connection types of Frame Structure

Connecting Different Kinds of Elements

Structural Symmetry

Loading Condition for

Underground Tunnel Modeling

Boundary Condition for Bored Pile

Subgrade Modeling Using Solid Elements

Bottom-up Method

Geometrical Modeling Method

Basic Tips of Geometrical Modeling Method

Modeling Method Using CAD Model

Plane Stress and Plane Strain Modeling

Modeling for Vessel Foundation

Foundation Analysis Programs

Linear & Nonlinear System Modeling

Isolation Plan with Expansion Joints

Global FE Model (Preliminary)

Structural Component for FE Modeling

and Analysis

Thermal Structural Analysis Using

Nonlinear Frictional Contact

Maximum Spacing of Expansion Joint

B.C. Effects in Thermal Structural Analysis

Constraint Equation

Application of Boundary Conditions

to Obtain Better FEA Results

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FEA Modeling & Analysis

FE Model Generation

Structural Analysis

for Component

Isolation Plan

Structural Analysis

with Symmetric

Boundary Condition

Example 1: Structure with Single

Component

Example 2: Structure with Multi

Components

Example 3: FE Analysis for Global

Structural Behavior

Example 5: Thermal Structural

Analysis Using Linear Horizontal

Supports

Example 4: Thermal Structural

Analysis Using Nonlinear

Frictional Contact

Example 6: Local Detail Thermal

Structural Analysis (Plane Strain)

Example 7: Evaluation of

Structural Integrity (Tower Crane

Foundation)

Example 8: Evaluation of Concrete

Crack (Equipment Foundation)

Example 9: Thermal Analysis

(Temperature Distribution)

Contents - cont.

Examples of

Finite Element Modeling & Analysis

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1. Finite Element Method (FEM)

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Purpose

To solve problems with complicated geometries, loadings, and material properties where analytical solutions cannot be obtained

To understand the physical behaviors of a complex object (strength, heat transfer capability, fluid flow, etc.)

To predict the performance and behavior of the design; to calculate the safety margin; and to identify the weakness of the design accurately

To identify the optimal design with confidence


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FEM

approximate


Fundamental Concepts

Many engineering phenomena can be expressed by “governing equations” and “boundary conditions”

Elastic problems

Thermal problems

Fluid flow

Electrostatics

etc.

Governing Equation

(Differential Equation)

𝐿 𝜙 +𝑓 = 0

Boundary Conditions

𝐵 𝜙 + 𝑔 = 0

𝑲 𝒖 = 𝑭

A Set of Simultaneous Algebraic Equation

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Fundamental Concepts – cont.


Property [K] Behavior {u} Action {F}

Elastic stiffness displacement force

Thermal conductivity temperature heat source

Fluid viscosity velocity body force

Electrostatic permittivity electric potential charge

𝑲 𝒖 = 𝑭 𝒖 = 𝑲 −𝟏 𝑭

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𝑲 𝒖 = 𝑭

: Stiffness matrix for one linear Spring element One type of degree of freedom Symmetric (forces are equal and opposite to equilibrium, -f1=f2) Singular (boundary condition is required, u1=0)



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𝑲 𝒖 = 𝑭



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Assembling Element Equations to Obtain Global Equation


𝑲 𝒖 = 𝑭

𝑲𝑬 𝒖𝑬 = 𝑭𝑬




1. Obtain the algebraic equations for each element2. Put all the element equations together

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Assembling Element Equations to Obtain Global Equation


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Real Structure

Simplified Physical Model

Discretization

FEM cuts a structure into several elements (pieces of the structure).

Then reconnects elements at “nodes” as if nodes were pins or drops of glue that hold elements together.

This process results in a set of simultaneous algebraic equations.

Discretized Model (mesh)


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Pre-Processing

Discretize Continuum (Modeling) Impose Boundary Conditions Impose External Forces

Solution (Internal Processing)

Find Element Stiffness Matrix Assemble Element Stiffness Matrix (System Stiffness Matrix) Solve Displacements Convert Displacement into Force, or Stress

Post-Processing

Sort, Print, and Plot Selected Results from Finite Element Solution


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Advantages

Can readily handle very complex geometry

Can handle a wide variety of engineering problems;Solid mechanics - Dynamics - Heat problems - Fluids - Electrostatic problems

Can handle complex loading;Nodal load, Element load, Time or frequency dependent loading

Disadvantages

The FEM obtains only "approximate" solutions.

The FEM has "inherent" errors.


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2. Pre-requisition for Structural Analysis

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Static Analysis Modal Analysis

Harmonic Analysis

Transient Dynamic Analysis

Spectrum Analysis

Buckling Analysis Explicit Dynamic Analysis

Available onlyin Linear Analysis ← Linear B.C. Required

Typical applications Drop tests Impact and Penetration

Types of Structural Analysis

𝑴 𝒖 + 𝑪 𝒖 + 𝑲 𝒖 = 𝑭(𝒕)

General Equation of Motion

𝑴 𝒖 + 𝑲 𝒖 = 𝟎

Linear Equation of Motion for Free, Un-damped Vibration


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Nonlinear Structural Analysis

Geometric Nonlinearities:If a structure experiences large deformations, its changing geometric configuration can cause nonlinear behavior.

Material Nonlinearities: A nonlinear stress-strain relationship, such as metal plasticity shown on the right, is another source of nonlinearities.

Boundary Condition (Contact) : “changing status” nonlinearity, where an abrupt change in stiffness may occur when bodies come into or out of contact with each other.

← compress only spring included


Source: ANSYS Mechanical Introduction to Structural Nonlinearities

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Element Types

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Element TypeTranslation Rotation

Required DataX Y Z X Y Z

Truss Yes Yes Yes Area

Beam Yes Yes Yes Yes Yes Yes Area

2D Solid Yes Yes

Membrane Yes Yes Yes Thickness

Plate Yes Yes Yes Yes* Yes* Thickness

Solid Yes Yes Yes


Degree of Freedom of Each Element Type

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Element Coordinate Systems of Shell and Beam Elements


Element Coordinate Systems

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Element Output Data

Sign Convention of Shell Element Forces

X Directional Stress due to Moment (Mx)

Source: STAAD.Pro – Technical Reference Manual

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Element Output Data – cont.

Three Directional Stresses of Solid Element

Solid element can simulate shear deformation and nonlinear stress distribution in thick members.

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Calculation of Design Moments

Shell Element

Solid Element

𝑴𝑼𝑿 = 𝑴𝑿 + 𝑨𝑩𝑺 𝑴𝑿𝒀 ∗ 𝑺𝑮𝑵(𝑴𝑿)


𝑴𝑼𝒀 = 𝑴𝒀 + 𝑨𝑩𝑺 𝑴𝑿𝒀 ∗ 𝑺𝑮𝑵(𝑴𝒀)

𝑴𝑼𝑿 = (𝒛𝒊 − 𝒛𝒄) × (𝝈𝑿𝒊 − 𝝈𝑿)𝑨𝒊

𝑴𝑼𝒀 = (𝒛𝒊 − 𝒛𝒄) × (𝝈𝒀𝒊 − 𝝈𝒀)𝑨𝒊 𝝈𝑿𝒊

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Nonlinear Behaviors of Real Structure

Concrete Cracked-elastic Stresses Stage Ultimate Stresses Stage

Euler-Bernoulli vs TimoshenkoShear Deformation

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3. Typical Boundary Conditions

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Moment ConnectionsShear Connections


Connection types of Frame Structure

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All connections have a certain amount of rigidity

Simple connections (A above) have some rigidity, but are assumed to be free to rotate

Partially-Restrained moment connections (B and C above) are designed to be semi-rigid

Fully-Restrained moment connections (D and E above) are designed to be fully rigid


Rigidity of Each Connection Type

Source: AISC Teaching Aids - Connections and Bracing Configurations

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Result Changes due to Boundary Conditions

Displacement (Y Direction) Moment (Z Direction)

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Connecting Different Kinds of Elements

Connecting Shell to Solid (No Moment Transferred) Connecting Beam to Shell (No Torque Transferred)

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Connecting beam element to plane elements: (a) no moment is transferred, (b) moment is transferred

Connecting Different Kinds of Elements – cont.


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Different Types of Structural Symmetry


Structural Symmetry

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Applied Structural Symmetry

Modelling a cubic block with two planes of symmetry Problem reduction using axes of symmetry applied to

a plate with a hole subjected to tensile force

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Applied Structural Symmetry – cont.

Simply supported symmetric beam structure Simply supported anti-symmetric beam structure

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Rigid Corner of Frame Structure

Coupling, Offset, Rigid Member, etc.

Source: Finite element design of concrete structures

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Constraint Equation


Every node tied together has the same value for degree of freedom

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Loading Condition for Underground Tunnel Modeling (Plus Dynamic Earth Pressure)

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Loading Condition for Underground Tunnel Modeling (Flooding)


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Result Changes due to Boundary Conditions

1. Fixed Condition 2. Vertical Springs3. Compression-only Vertical Springs

3

2

1


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Horizontal Boundary Condition for Pile Modeling

FE Model and Distribution of Subgrade Reaction Modulus for Horizontal Force at Pile Head

Piles can be modelled by linear-elastic supported beamelements.

The bedding modulus ks and the stiffness of the horizontal springs may vary along the length of the pile and its circumference.

Exponent n should be chosen as follows;

n Soil Condition

0 cohesive soil under small to medium loads

0.5medium cohesive soil and non-cohesive soil above ground water level

1non-cohesive soil below ground water level or under greater loads

1.5 to 2 loose non-cohesive soil under very high loads

𝒌𝒔 𝒛 = 𝒌𝒔 × 𝒅 × (𝒛/𝒅)𝒏

Source: Finite element design of concrete structures

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Influence on Analysis Results by Stiffness of Vertical Spring

Bending moment distribution in pile(horizontal load: 870 kN at column head)

Horizontal deformation of pile (horizontal load: 870 kN at column head)

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Pile Model with Strut-and-Tie - Foundation of Bridge Pier


Strut-and-tie Model for Pile Cap

Strut-and-tie Model for Pile Cap

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Subgrade Modeling Using Solid Elements

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X

Y

Z

B.C.:

SYMMETRIC

B.C.: UY=0

B.C.: UX=0

SUBGRADE ELEMENTS

CONCRETE ELEMENTS

L2 = 7.5 m

L1 = 5.0 m

FE Model for Parametric Study

Case Contact B.C. Load σx_top σx_bottom Moment Axial Force Remark

1 None Thermal ≒0 ≒0 ≒0 ≒0 No stress w/o constraint

2 FixedGravity

Thermal-2683.06 2673.64 446.39 ≒0

w/o Subgrade Elements

No axial force

3 FrictionGravity

Thermal-1310.53 1284.77 216.28 12.88

4 w/o FrictionGravity

Thermal-1316.63 1317.22 219.49 ≒0 No axial force

5 MergedGravity

Thermal-2129.38 1730.25 321.64 -199.57

Parametric Study Results



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Case 1. w/o gravity, w/o Friction (σx) Case 2. w/ gravity, w/o Friction (σx, Fixed B.C.)

Case 3. w/ gravity, w/ Friction (σx) Case 4. w/ gravity, w/o Friction (σx) Case 5. w/ gravity, shared nodes on the interface surface of soil and concrete (σx)



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4. Element Mesh Generation

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Make nodes

Build elements by assigning connectivity

Apply boundary conditions and loads


Bottom-up Method for Element Generation

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Geometrical Modeling Method for Element Generation

Geometrical Modeling

(a) Physical Geometry of Structural Parts

(b) Geometry Created in FE Model

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Basic Tips of Geometrical Modeling Method

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It is important to remember that a finite element solution is an approximation:• CAD geometry is an idealization of the physical model.• The mesh is a combination of discreet “elements” representing the geometry.• The accuracy of answers is determined by various factors, one of which is the mesh density.


Modeling Method Using CAD Model

3D CAD Model Finite Element Model

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Modeling Method Using CAD Model – cont.

3D CAD Model Finite Element Model

Left View Right View

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Navisworks Screenshot of Frameworks Model (Part of PDS Model)

XY

Z

3D Isometric View of 3D Frame Model (STAAD.Pro v8i)

3D Rendered Isometric Views (STAAD.Pro v8i)



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Automated Structural Analysis System

Build 3D CAD Model

Convert 3D CAD Model to Finite Element Model

Generate Input Data Based on Load Database

Under Development of Different Modules Specialized for Each Structure



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3D Shell Element Mesh Imported into FEA Program

Steel Concrete Composite Column

Members

Steel Beam and Girder Members



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5. FE Analysis Boundary Based on Structural Behavior

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Plane Stress / Plane Strain Problems

Plane strain problems: (a) dam subjected to horizontal loading (b) pipe subjected to a vertical load

Plane stress problems: (a) plate with hole; (b) plate with fillet

Source: A FIRST COURSE IN THE FINITE ELEMENT METHOD (Daryl L. Logan)

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Plane Stress / Plane Strain Problems – cont.

Plot of minimum principal stress with largest absolute value of 1.86 MPa located on back side of dam subjected to both hydrostatic and self-weight loading


Mohr’s Circle for Plane Strain

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Seepage Analysis – Potential ProblemBoundary Condition (left) and Hydraulic Head Contour (right)



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Seepage Analysis – Potential ProblemFlow Velocity Vector (left) and Equipotential Lines (right)


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Modeling for Vessel Foundation

Shell Foundation Model w/o Pedestal Stiffness

Shell Foundation Model with Solid Pedestal

Solid Foundation Model with Solid Pedestal

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+

Compression-only Soil Spring

Fixed B.C. for Shell Foundation(or Hinge B.C. for Solid Foundation)

Shell Elements Foundation

Beam Elements Support Frame

Linear System for Dynamic Analysis

Nonlinear System for Static Analysis


Rigid Link

Vessel Mass

Linear & Nonlinear System Modeling

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Linear & Nonlinear System Modeling – cont. (Super-structure)

Stack +AB (Modal-05)

TYPE MASS (1000 kg)

1 680.891 Stack Shell

2 20.077 Stack Beam

3 524794.000 AB Shell

4 1618.620 AB Beam

527113.588

Decoupling Criteria for Subsystems U.S. NRC SPR 3.7.2

If Rm < 0.01, decoupling can be done for any Rf.

If 0.01 < Rm < 0.1, decoupling can be done if 0.8 > Rf > 1.25.

If Rm > 0.1, a subsystem model should be included in the primary system model.

Rm = Total mass of supported subsystem / Dominant mass of supporting system

Rf = Total mass of supported subsystem / Dominant mass of supporting system

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Linear & Nonlinear System Modeling – cont. (Foundation)

Modeling Concept

Selected Solid Elements to Consider Various Thickness Changes

Applied Compression-only Spring for Simulating Uplifting

Coupled Super-structure with Zero Density to Use Its Stiffness

Nonlinear System for Static Analysis

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6. Examples of B.C. Applications

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Analysis Models and B.C. ApplicationBased on Structural Behaviors


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Global FE Model (Preliminary)

To Check Stability and Structural Behavior

Compress-only Springs Used to Consider Buoyancy

Loading Condition: Self-weight, Soil Pressure, Buoyancy


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2D Model (2 EA)

Plane Strain Behavior

3D Model (9 EA)

3D Structural Behavior

Structural Component for FE Modeling and Analysis (Design Purpose)


Each structural component should be isolated to match actual structural behaviors to the assumed in splitting the entire structure. In this case, expansion joints are arranged for the purpose.

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Thermal Structural Analysis Using Nonlinear Frictional Contact


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Thermal Structural Analysis to Determine

Maximum Spacing between Expansion Joints

Analysis Boundary: Separated Bay (Orange-colored)

Seasonal Change in Temperature (Case 1) :

T0 (ref. temp)=27.5℃, △T= -22.5℃ (Ambient Air)

Daily Change in Temperature (Case 2):

T0 (ref. temp)=40.0℃, △T= +22.5℃ (Solar Radiation)

Boundary Condition:

Accounts for Friction Effects between Concrete and

Subgrade

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Type of Building Outside Temperature Variations Maximum Joint Spacing (ft)

HeatedUp to 70°F

Above 70°F

600 (182.88 m)

400-500 (121.92 ~152.4 m)

UnheatedUp to 70°F

Above 70°F

300 (91.44 m)

200 (60.96 m)

Mark Fintel, Section 4.10.2, "Spacing of Expansion Joints", Handbook of Concrete

Engineering, pp. 129-130.

ACI Report: Building Movements and Joints, EB086.01B.Buildings of more than 600 ft (183 m) have been constructed and performed satisfactorily without expansion joints. The possible need for thermal expansion joints in long buildings may be determined initially using the empirical approach described in the following section. Previously developed empirical rules for expansion joint spacing are not necessarily compatible with modern construction. Therefore, effects of thermal and other volume changes should be determined as part of the structural analysis. If results of the empirical approach indicate an expansion joint may be needed, a more comprehensive analysis can be done to determine if use of expansion joints can be avoided.


Maximum Spacing between Expansion Joints

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Thermal Loading Condition due to Solar Radiation

Temperature Distribution through Cross Section of Aeration Channel

Temperature Distribution through Aeration Channel

Design Temperature Condition

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Nonlinear Frictional Contact between Concrete and Subgrade Parts

Nonlinear Contact Boundary Condition

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Model Verification

Total Deformation under Thermal Load only (ISO View)

This condition cannot occur in the real loading cases under gravity, but it has to be checked to verify nonlinear contact boundary condition.

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Analysis Results

Total Deformation under Thermal Load and Self-weight

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Normal Stress in Global Z Direction

Normal Stress in Global Z Direction (Upside Down)

Analysis Results