Chapter
1.6. Basic Steps of Programming the Direct Stiffness Method
Chapter Two: Plane Trusses
2.2. Overview of the Plane Truss
2.3. Vectors of End-Actions of Plane Truss Element
2.4. End-Translation Vectors of Plane Truss Element
2.5. Transformation Matrix of Plane Truss Element
2.5.1. Application—Transformation Matrices of Truss Member
2.6. Total Local Stiffness Matrix of Plane Truss Element
2.6.1. Application—Local Truss Stiffness Matrices
2.7. Total Global Stiffness Matrix of Plane Truss Element
2.7.1. Application—Global Stiffness Matrices of Plane Truss Elements
2.8. Vectors of Nodal-Forces and Nodal-Translations of Plane Truss
2.8.1. Application—Total Nodal Forces and Translations
2.9. Global Stiffness Matrix of Plane Trusses
2.9.1. Application—Global Stiffness Matrix of Plane Truss
2.10. Modification of Global Stiffness Matrix Due to Support Conditions—Reordering Matrix
2.10.1. Application—Modification of Plane Truss Stiffness Equation Due to Reordering—Computation of Unknown Quantities
2.11. Modification of Global Stiffness Matrix of a Plane Truss Due to Inclined and Elastic Supports
2.11.3. Application—Modification of Stiffness Equation of Plane Truss Due to Inclined or/and Elastic Support—Computation ...
2.12. Plane Truss Subjected to Member Loading
2.12.1. Application—Analysis of Plane Truss Subjected to Member Loading—Computation of Unknown Quantities
2.13. Stress Resultants of Plane Truss Members
2.13.1. Application—Computation of Stress Resultants of Plane Truss Elements
Chapter Three: Plane Frames
3.2. Overview of the Plane Frame
3.3. Vectors of End-Actions of Plane Frame Element
3.4. End-Displacement Vector of Plane Frame Element
3.5. Transformation Matrix of Plane Frame Element
3.5.1. Application—Transformation Matrices of Plane Frame Member
3.6. Total Local Stiffness Matrix of Plane Frame Element
3.6.1. Application-Local Frame Stiffness Matrices
3.7. Total Global Stiffness Matrix of Plane Frame Element
3.7.1. Application—Global Stiffness Matrices of Plane Frame Elements
3.8. Vectors of Nodal-Forces and Nodal-Displacements of Plane Frame
3.8.1. Application—Total Nodal Forces and Displacements of Frame
3.9. Global Stiffness Matrix of Plane Frames
3.9.1. Application—Global stiffness matrix of plane frame
3.10. Modification of Global Stiffness Matrix Due to Support Conditions—Reordering Matrix
3.10.1. Application—Modification of Plane Frame Stiffness Equation Due to Reordering—Computation of Unknown Quantities
3.11. Modification of Global Stiffness Matrix of a Plane Frame Due to Inclined And Elastic Supports
3.11.3. Application—Modification of Stiffness Equation of a Plane Frame Due to Inclined or/and Elastic Support—Computatio ...
3.12. Plane Frame Subjected to Member Loading
3.12.1. Application—Analysis of Plane Frame Subjected to Member Loading—Computation of Unknown Quantities
3.13. Stress Resultants of Plane Frame Members
3.13.1. Application—Computation of Stress Resultants of Plane Frame Members
Chapter Four: Spatial Trusses
4.2. Overview of Spatial Truss
4.3. Vectors of End-Actions of Spatial Truss Element
4.4. End-Translation Vectors of Spatial Truss Element
4.5. Transformation Matrix of Spatial Truss Element
4.5.1. Application—Transformation Matrices of Truss Members
4.6. Total Local Stiffness Matrix of Spatial Truss Element
4.6.1. Application—Local Stiffness Matrices of Spatial Truss Members
4.7. Total Global Stiffness Matrix of Spatial Truss Element
4.7.1. Application—Global Stiffness Matrices of Spatial Truss Elements
4.8. Vectors of Nodal-Forces and Nodal-Translations of Spatial Truss
4.8.1. Application—Vectors of Total Nodal Forces and Translations of Truss
4.9. Global Stiffness Matrix of Spatial Trusses
4.9.1. Application—Global Stiffness Matrix of Spatial Truss
4.10. Modification of Global Stiffness Matrix Due to Support Conditions—Reordering Matrix
4.10.1. Application—Modification of Spatial Truss Stiffness Equation Due to Reordering—Computation of Unknown Quantities
4.11. Internal Forces of Spatial Truss Members
4.11.1. Application—Computation of Internal Stress Resultants of Spatial Truss Elements
Chapter Five: Spatial Frames
5.2. Overview of Spatial Frames
5.3. Vectors of End-Actions and End-Displacements of Spatial Frame Element
5.4. Transformation Matrix of Spatial Frame Element
5.4.1. Basic Transformation Matrix
5.4.2. Transformation Matrix with Special Orientation
5.4.3. Transformation Matrix for Special Auxiliary Point
5.4.4. Transformation Matrix of End-Components of a Spatial Frame Element
5.4.5. Formulation of Transformation Matrices of Elements of Other Type of Skeletal Structures
5.4.6. Application—Transformation Matrices of Space Frame Members
5.5. Local Stiffness Matrix of Spatial Frame Element
5.5.1. Formulation of Stiffness Matrix—Stiffness Terms
5.5.2. Stiffness Matrix of Spatial Frame Member of Constant Cross Section
5.5.3. Formulation of Local Stiffness Matrices of Members of all Other Types of Skeletal Structures
5.5.4. Application—Local Stiffness Matrices of Spatial Frame Members
5.6. Global Stiffness Matrix of Spatial Frame Element
5.6.1. Application—Global Stiffness Matrices of Spatial Frame Elements
5.7. Vectors of Nodal-Actions and Nodal-Displacements of Spatial Frame
5.7.1. Application—Vectors of Total Nodal Actions and Displacements
5.8. Global Stiffness Matrix of Spatial Frame
5.8.1. Application—Global Stiffness Matrix of Spatial Frame
5.9. Modification of Global Stiffness Matrix of Spatial Frame Due to Support Conditions—Reordering Matrix
5.9.1. Application—Modification of Spatial Frame Stiffness Equation Due to Reordering—Computation of Unknown Quantities
5.10. Internal Actions of Spatial Frame Members
5.10.1. Application—Computation of Internal Stress Resultants of Spatial Frame Elements
5.11. Application—Analysis of Grid Structure
5.11.1. Numbering of Nodes and Members, Global, and Local Systems of Axes, Nodal Degrees of Freedom
5.11.2. Formulation of Local Stiffness Matrices of Grid Structure Members
5.11.3. Formulation of Transformation Matrices of Grid Elements
5.11.4. Formulation of Global Stiffness Matrices of Grid Elements
5.11.5. Assembly of the Global Stiffness Matrix of the Grid Structure
5.11.6. Reordering of the Global Stiffness Matrix of the Grid
5.11.7. Computation of Restrained Structure Nodal Actions
5.11.8. Computation of Total Nodal Actions and Displacement Vectors of Grid Structure
5.11.9. Computation of Unknown Nodal Displacements and Support Reactions
5.11.10. Computation of End-Actions and Internal Actions Diagrams of Grid Members
Chapter Six: Rigid Joints
6.2. Kinematic Relations and Equivalent Actions Betweeen two Points of a Rigid Body Plane
6.3. Rigid Joints in Plane Framed Structure
6.4. Analysis of Plane Frame With Rigid Joints
6.4.1. Numbering of Nodes and Members, Global, and Local Systems of Axes, Nodal Degrees of Freedom (dof)
6.4.2. Formulation of Local Stiffness Matrices of the Members
6.4.3. Modification of Local Stiffness Matrix of Member 2 Due to Rigid Joint
6.4.4. Formulation of Transformation Matrices of the Members
6.4.5. Formulation of Global Stiffness Matrices of the Members
6.4.6. Assembly of the Global Stiffness Matrix of the Entire Frame
6.4.7. Modification of the Global Stiffness Matrix of the Frame Due to Inclined Support
6.4.8. Reordering of the Global Stiffness Matrix of the Frame
6.4.9. Computation of Restrained Structure Nodal Actions
6.4.10. Computation of Total Nodal Actions and Displacement Vectors
6.4.11. Computation of Unknown Nodal Displacements and Support Reactions
6.4.12. Computation of End-Actions and Internal Actions Diagrams of Frame Members
6.5. Kinematic Relations and Equivalent Actions Between two Points of Space Rigid Structure
6.6. Rigid Joints in Space Frame Element
Chapter Seven: Internal Releases—Method of Combined Nodes
7.2. Degrees of Freedom of Combined Nodes
7.3. Assembly of Total Global Stiffness Matrix With Combined Nodes
7.4. Computation of Nodal Actions of Restrained and Equivalent Structures With Combined Nodes
7.6. Application—Analysis of Plane Frame With Combined Nodes
7.6.1. Numbering of Nodes and Members, Global and Local Systems of Axes, Nodal Degrees of Freedom
7.6.2. Formulation of Local Stiffness Matrices of the Members
7.6.3. Formulation of Transformation Matrices of the Members
7.6.4. Formulation of Global Stiffness Matrices of the Members
7.6.5. Assembly of the Global Stiffness Matrix of the Entire Frame
7.6.6. Reordering of the Global Stiffness Matrix of the Frame
7.6.7. Computation of Restrained Structure Nodal Actions
7.6.8. Computation of Total Nodal Actions and Displacement Vectors
7.6.9. Computation of Unknown Nodal Displacements and Support Reactions
7.6.10. Computation of End-Actions and Internal Action Diagrams of Frame Members
Chapter Eight: Internal Hinges—Modified Stiffness Matrix Method
8.2. Modified Stiffness Matrices
8.2.1. Elimination of the Rotation dof of the End k of a Plane Beam Element (P2)
8.2.2. Elimination of the Transverse dof (Transverse Release) of the End k of a Plane Beam Element
8.3. Modified Matrices and Internal Releases
8.4. Restrained Actions—Equivalent Actions
8.4.1. Restrained Actions of a Modified Element Subjected to Member Loading
8.4.2. Restrained Actions of a Modified Element Subjected to Loading Applied at the Internal Release
8.4.3. Restrained Actions of a Modified Element Subjected to Support Displacement
8.5. Application—Analysis of Plane Frame With the Modified Stiffness Matrix Method
8.5.1. Numbering of Nodes and Members, Global and Local Systems of Axes, and Nodal Degrees of Freedom
8.5.2. Formulation of Local Stiffness Matrices, Transformation Matrices and Global Stiffness Matrices of the Frame Members
8.5.3. Modification of the Global Stiffness Matrix of Frame Member
8.5.4. Assembly of the Global Stiffness Matrix of the Entire Frame
8.5.5. Reordering of the Global Stiffness Matrix of the Frame
8.5.6. Computation of Restrained Structure Nodal Actions
8.5.7. Computation of Total Nodal Actions and Displacement Vectors
8.5.8. Computation of Unknown Nodal Displacements and Support Reactions
8.5.9. Computation of Eliminated Displacements Along Global dof 7,8
8.5.10. Computation of End-Actions and Internal Actions Diagrams of Frame Members
Chapter Nine: Static Condensation Method
9.2. Physical Interpretation of Static Condensation
9.3. Qualitative Examination of the Stiffness Coefficients of a Hyperelement
9.4. Stiffness Matrix and Restrained Actions With Elastic Hinge
9.5. Application—Different Modeling Considerations of a Plane Frame Structure
9.5.1. Numbering of Nodes and Members, Global and Local Systems of Axes, Nodal Degrees of Freedom
9.5.2. Formulation of Local Stiffness Matrices of the Members
9.5.3. Formulation of Transformation Matrices of the Members
9.5.4. Formulation of Global Stiffness Matrices of the Frame Members
9.5.5. Assembly of the Global Stiffness Matrix of the Entire Frame According to Modeling Consideration (i)
9.5.6. Reordering of the Global Stiffness Matrix of the Frame
9.5.7. Computation of Restrained Structure Nodal Actions
9.5.8. Computation of Total Nodal Actions and Nodal Displacement Vectors
9.5.9. Computation of Unknown Nodal Displacements and Support Reactions
9.5.10. Computation of End-Actions and Internal Action Diagrams of Frame Members
9.5.11. Assembly of the Global Stiffness Matrix of the Entire Frame According to Modeling Consideration (ii)
9.5.12. Reordering of the Global Stiffness Matrix of the Frame
9.5.13. Computation of Restrained Structure Nodal Actions
9.5.14. Computation of Total Nodal Actions and Displacement Vectors
9.5.15. Computation of Unknown Nodal Displacements and Support Reactions
9.5.16. Computation of Eliminated Displacement Along Global dof 5
9.5.17. Computation of End-Actions and Internal Actions Diagrams of Frame Members
9.5.18. Assembly of the Global Stiffness Matrix of the Entire Frame According to Modeling Consideration (iii)
9.5.19. Reordering of the Global Stiffness Matrix of the Frame
9.5.20. Computation of Restrained Structure Nodal Actions
9.5.21. Computation of Total Nodal Actions and Nodal Displacement Vectors
9.5.22. Computation of Unknown Nodal Displacements and Support Reactions
9.5.23. Computation of Eliminated Displacement Along Global dof 5, 6
9.5.24. Computation of End-Actions and Internal Actions Diagrams of Frame Members
9.5.25. Consideration (iv): Static Condensation of All Global dof of Node 2—Formulation of Hyperelement 1-2-3
9.5.26. Reordering of the Global Stiffness Matrix of the Hyperelement
9.5.27. Computation of Nodal Actions and Nodal Displacement Vectors of the Hyperelement
9.5.28. Computation of Nodal Displacements and Support Reactions of the Hyperelement
9.5.29. Computation of Eliminated Internal Global Displacements of the Hyperelement (Node 2)
9.5.30. Computation of End-Actions and Internal Actions Diagrams of Frame Members
Chapter Ten: Elements of Variable Cross Section
10.2. Stiffness Matrix—Analytic Evaluation
10.3. Stiffness Matrix—Approximate Computation
10.4. Restrained Actions—Analytic Computation
10.5. Restrained Actions—Approximate Computation
10.6. Application—Analysis of An Element of Variable Cross Section
10.6.1. Numbering of Nodes and Members, Global and Local Systems of Axes, and Nodal Degrees of Freedom
10.6.2. Formulation of Local Stiffness Matrices of the Members
10.6.3. Formulation of Transformation Matrices of the Members
10.6.4. Formulation of Global Stiffness Matrices of the Members
10.6.5. Modification of the Global Stiffness Matrix of Member 2 Due to the Hinge at Node 3 [Modeling Consideration (i)]
10.6.6. Assembly of the Global Stiffness Matrix of the Structure
10.6.7. Computation of Restrained Structure End-Actions
10.6.8. Modification of the Global Vector of the End-Actions of Member 2 of the Restrained Structure, Due to Elimination ...
10.6.9. Formulation of the Global Equilibrium Equation of the Structure—Computation of Displacements, Reactions and Stres ...
10.6.10. Computation of Eliminated Rotation of End k of Member 2
10.6.11. Assembly of the Stiffness Matrix of the Hyper Element [Modeling Consideration (ii)]
10.6.12. Static Condensation of the Global Stiffness Matrix of the Hyperelement Due to Elimination of the Internal dof—De ...
10.6.13. Computation of the Vector of Total Nodal Actions and Displacements of the Hyperelement
10.6.14. Computation of the Vector of the End-Actions of the Hyperelement With Variable Cross Section Due to Elimination ...
10.6.15. Formulation of the Equilibrium Equation of the Element of Variable Cross Section—Computation of Displacements, R ...
10.6.16. Computation of Eliminated Displacements Along the Internal dof
Chapter Eleven: The Method of Substructures
11.2. Presentation of Method of Substructures: A Plane Frame Test Case
11.3. The Method of Substructures: A Plane Truss Test Case
11.4. The Method of Substructures: A Plane Truss Numerical Application
11.4.1. Subdivision to Substructures, Free Body Diagrams of Substructures
11.4.2. Formulation of the Reordered Stiffness Matrices of the Substructures (Hyperelements)
11.4.3. Static Condensation of the Internal dof of the Substructures
11.4.4. Composition of the Stiffness Matrix With Respect to the Hypernodes of the Structure
11.4.5. Computation of the Actions of the Hypernodes of the Equivalent Structure
11.4.6. Assembly and Solution of the Stiffness Equation With Respect to the Hypernodes
11.5. The Method of Substructures: Numerical Application on a Hybrid Structure
11.5.1. Subdivision to Substructures, Free Body Diagrams of Substructures
11.5.2. Formulation of the Reordered Stiffness Matrices of the Substructures—Hyperelements
11.5.2.1. Substructure A—Numbering of Nodes and Members, Global and Local Systems of Axes, Nodal Degrees of Freedom
11.5.2.2. Substructure A—Formulation of Local Stiffness Matrices of the Members
11.5.2.3. Substructure A—Formulation of Members Transformation Matrices
11.5.2.4. Substructure A—Formulation of Global Stiffness Matrices of the Substructure's Members
11.5.2.5. Substructure A—Assembly of the Global Stiffness Matrix of the Substructure—Hyperelement
11.5.2.6. Substructure A—Modification of the Global Stiffness Matrix Due to Inclined Support
11.5.2.7. Substructure A—Reordering of the Global Stiffness Matrix of the Substructure—Hyperelement
11.5.2.8. Substructure A—Formulation of the Reordered Stiffness Matrix of the Substructure—Hyperelement
11.5.2.9. Substructure B—Formulation of the Reordered Stiffness Matrix of the Substructure—Hyperelement
11.5.2.10. Substructure C—Formulation of the Reordered Stiffness Matrix of the Substructure—Hyperelement
11.5.3. Static Condensation of Internal dof of Substructures
11.5.4. Stiffness Matrix Assembly With Respect to the Structure's Hypernodes
11.5.5. Computation of the Equivalent Actions of Structure Hypernodes
11.5.5.1. Substructure —Computation of Restrained Substructure Nodal Actions
11.5.5.2. Substructure A—Computation of Total Nodal Actions and Displacement Vectors of Equivalent Substructure—Hyperelement
11.5.5.3. Substructure B—Computation of Total Nodal Actions and Displacement Vectors of Equivalent Substructure—Hyperelement
11.5.5.4. Substructure C—Computation of Total Nodal Actions and Nodal Displacement Vectors of Equivalent Substructure—Hyp ...
11.5.6. Static Condensation of Nodal Actions of Equivalent Structure's Hypernodes Due to Elimination of Internal dof of S ...
11.5.7. Formulation of the Final Stiffness Matrix Equation With Respect to the Displacements of the Hypernode of the Plan ...
11.5.8. Remaining Steps for the Entire Solution of the Plane Hybrid Structure
Chapter Twelve: Programming of Direct Stiffness Method—PFrameMatlab Program
12.1. Basic Steps of Programming
12.3. Individual Program Files
File: PFrameMatlab.m (Main executable program)
File: PFrameAssignElementProperties.m
File: PFrameElementLength.m
File: PFrameT1ElementStiffness.m
File: PFrameT2ElementStiffness.m
File: PFrameT3ElementStiffness.m
File: PFrameT4ElementStiffness.m
File: PFrameElementTransformation.m
12.4. Application—Plane Frame Analysis
Input data file: model1.dat
Graphical output of the structures
Output file: model1.dat.out
12.5. Application—Hybrid Plane Frame Analysis
Input data file: model2.dat
Graphical output of the structure
Output file: model2.dat.out
Appendix A—Tensor Calculus
A.2.1. First-Order Tensors
A.2.2. Second-Order Tensors
A.3. Transformation of Tensors
A.3.1. Transformation of a Vector (First-Order Tensor)
A.3.2. Transformation of the Second-Order Tensor
Matrix Methods for Advanced Structural Analysis
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