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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
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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
1. Finite Element Method (FEM)
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FEM
approximate
1. Finite Element Method (FEM)
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.
1. Finite Element Method (FEM)
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)
1. Finite Element Method (FEM)
Fundamental Concepts – cont.
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𝑲 𝒖 = 𝑭
1. Finite Element Method (FEM)
Fundamental Concepts – cont.
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Assembling Element Equations to Obtain Global Equation
1. Finite Element Method (FEM)
𝑲 𝒖 = 𝑭
𝑲𝑬 𝒖𝑬 = 𝑭𝑬
𝑲𝑬 𝒖𝑬 = 𝑭𝑬
𝑲𝑬 𝒖𝑬 = 𝑭𝑬
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
1. Finite Element Method (FEM)
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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)
1. Finite Element Method (FEM)
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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
1. Finite Element Method (FEM)
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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.
1. Finite Element Method (FEM)
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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
2. Pre-requisition for Structural Analysis
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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
2. Pre-requisition for Structural Analysis
Source: ANSYS Mechanical Introduction to Structural Nonlinearities
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2. Pre-requisition for Structural Analysis
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
2. Pre-requisition for Structural Analysis
Degree of Freedom of Each Element Type
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Element Coordinate Systems of Shell and Beam Elements
2. Pre-requisition for Structural Analysis
Element Coordinate Systems
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2. Pre-requisition for Structural Analysis
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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2. Pre-requisition for Structural Analysis
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)
2. Pre-requisition for Structural Analysis
𝑴𝑼𝒀 = 𝑴𝒀 + 𝑨𝑩𝑺 𝑴𝑿𝒀 ∗ 𝑺𝑮𝑵(𝑴𝒀)
𝑴𝑼𝑿 = (𝒛𝒊 − 𝒛𝒄) × (𝝈𝑿𝒊 − 𝝈𝑿)𝑨𝒊
𝑴𝑼𝒀 = (𝒛𝒊 − 𝒛𝒄) × (𝝈𝒀𝒊 − 𝝈𝒀)𝑨𝒊 𝝈𝑿𝒊
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2. Pre-requisition for Structural Analysis
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
3. Typical Boundary Conditions
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
3. Typical Boundary Conditions
Rigidity of Each Connection Type
Source: AISC Teaching Aids - Connections and Bracing Configurations
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3. Typical Boundary Conditions
Result Changes due to Boundary Conditions
Displacement (Y Direction) Moment (Z Direction)
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3. Typical Boundary Conditions
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.
3. Typical Boundary Conditions
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Different Types of Structural Symmetry
3. Typical Boundary Conditions
Structural Symmetry
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3. Typical Boundary Conditions
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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3. Typical Boundary Conditions
Applied Structural Symmetry – cont.
Simply supported symmetric beam structure Simply supported anti-symmetric beam structure
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3. Typical Boundary Conditions
Rigid Corner of Frame Structure
Coupling, Offset, Rigid Member, etc.
Source: Finite element design of concrete structures
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Constraint Equation
3. Typical Boundary Conditions
Every node tied together has the same value for degree of freedom
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3. Typical Boundary Conditions
Loading Condition for Underground Tunnel Modeling (Plus Dynamic Earth Pressure)
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Loading Condition for Underground Tunnel Modeling (Flooding)
3. Typical Boundary Conditions
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Result Changes due to Boundary Conditions
1. Fixed Condition 2. Vertical Springs3. Compression-only Vertical Springs
3
2
1
3. Typical Boundary Conditions
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3. Typical Boundary Conditions
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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3. Typical Boundary Conditions
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
3. Typical Boundary Conditions
Strut-and-tie Model for Pile Cap
Strut-and-tie Model for Pile Cap
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3. Typical Boundary Conditions
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
3. Typical Boundary Conditions
B.C. Effects in Thermal Structural Analysis
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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)
3. Typical Boundary Conditions
B.C. Effects in Thermal Structural Analysis
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4. Element Mesh Generation
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Make nodes
Build elements by assigning connectivity
Apply boundary conditions and loads
4. Element Mesh Generation
Bottom-up Method for Element Generation
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4. Element Mesh Generation
Geometrical Modeling Method for Element Generation
Geometrical Modeling
(a) Physical Geometry of Structural Parts
(b) Geometry Created in FE Model
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4. Element Mesh Generation
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.
4. Element Mesh Generation
Modeling Method Using CAD Model
3D CAD Model Finite Element Model
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4. Element Mesh Generation
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)
4. Element Mesh Generation
Modeling Method Using CAD Model – cont.
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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
4. Element Mesh Generation
Modeling Method Using CAD Model – cont.
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3D Shell Element Mesh Imported into FEA Program
Steel Concrete Composite Column
Members
Steel Beam and Girder Members
4. Element Mesh Generation
Modeling Method Using CAD Model – cont.
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5. FE Analysis Boundary Based on Structural Behavior
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5. FE Analysis Boundary Based on Structural Behavior
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
5. FE Analysis Boundary Based on Structural Behavior
Mohr’s Circle for Plane Strain
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Seepage Analysis – Potential ProblemBoundary Condition (left) and Hydraulic Head Contour (right)
5. FE Analysis Boundary Based on Structural Behavior
Plane Stress / Plane Strain Problems – cont.
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5. FE Analysis Boundary Based on Structural Behavior
Seepage Analysis – Potential ProblemFlow Velocity Vector (left) and Equipotential Lines (right)
Plane Stress / Plane Strain Problems – cont.
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5. FE Analysis Boundary Based on Structural Behavior
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
5. FE Analysis Boundary Based on Structural Behavior
Rigid Link
Vessel Mass
Linear & Nonlinear System Modeling
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5. FE Analysis Boundary Based on Structural Behavior
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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5. FE Analysis Boundary Based on Structural Behavior
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
6. Examples of B.C. Applications
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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
6. Examples of B.C. Applications
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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)
6. Examples of B.C. Applications
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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6. Examples of B.C. Applications
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Thermal Structural Analysis Using Nonlinear Frictional Contact
6. Examples of B.C. Applications
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6. Examples of B.C. Applications
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.
6. Examples of B.C. Applications
Maximum Spacing between Expansion Joints
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6. Examples of B.C. Applications
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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6. Examples of B.C. Applications
Nonlinear Frictional Contact between Concrete and Subgrade Parts
Nonlinear Contact Boundary Condition
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6. Examples of B.C. Applications
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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6. Examples of B.C. Applications
Analysis Results
Total Deformation under Thermal Load and Self-weight
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6. Examples of B.C. Applications
Normal Stress in Global Z Direction
Normal Stress in Global Z Direction (Upside Down)
Analysis Results
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