lecture01-module 1_introduction and matrix algebra
TRANSCRIPT
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Matrix Structural AnalysisMatrix Structural Analysis CIVIL 517CIVIL 517
Dr. Usama EbeadDr. Usama EbeadCivil and Environmental Engineering DepartmentCivil and Environmental Engineering DepartmentUnited Arab Emirates UniversityUnited Arab Emirates University
ukQMember end force vector
Member end displacement vector
Member stiffness matrix in the local coordinate system
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Textbook
Parts of the material used in this course are obtained from:
Aslam Kasimali, Matrix Analysis of Structures, Copyright 1999,
Brooks/Cole Publishing Company
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Outline of Module 1
Historical Background
Classical, Matrix, and Finite Element Methods of Structural
Analysis
Classification of Framed Structures
Analytical Models
Fundamental Relationships for Structural Analysis
Linear versus Nonlinear Analysis
Software
Linear Algebra
Summary
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Historical Background
The theoretical foundation for matrix methods of structural analysis was laid by James Maxwell and George ManyJames Maxwell developed the consistent deformations method (the basis for the flexibility method)
and
George Many developed the slope deflection method (the basis for the stiffness method)
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Classical, Matrix, and Finite Element Methods of Structural Analysis
Classical methods:
Provide understanding of the structural behavior
Limited Requires hand calculations
Ex. Moment distribution method, conjugate beam method, consistent deformations method, slope-deflections method, etc.
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Matrix methods:
Developed and suitable for computer implementation
SystematicGeneral
Ex. Flexibility Method (Generalization of Consistent Deformations Method)
Stiffness Method (Originated from the classical Slope-Deflections Method)
Classical, Matrix, and Finite Element Methods of Structural Analysis
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Finite Element Method:
Originated as an extension of Matrix Analysis
Can be applied to structures and solids of any shape
The force-displacement relations are derived by work-energy principles from assumed displacement or stress function
(Such relations in Matrix Analysis are based on exact solution of underlying differential equations
Classical, Matrix, and Finite Element Methods of Structural Analysis
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Classification of Framed Structures
Plane Trusses
Beams
Plane Frames
Space Trusses
Grids
Space Frames
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Plane Trusses
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Beams
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Plane Frames
2 kN/m
6 m
8 m
2 kN/m
10 m
8 m
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Grids
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Space Frames
z
y
x
w
P
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Analytical Models
The most important step in the analysis is to establish the “as accurate as practically possible” Analytical model
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Line Diagram - Example 1
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Line Diagram – Example 2
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Line Diagram - Example 3
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Line Diagram - Example 4
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Fundamental Equations for Structural Analysis
Equilibrium Equations
Compatibility Conditions
Constitutive Relations
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Equilibrium Equations
0M
0F
0F
y
x x
y
z
F 0
F 0
F 0
x
y
z
M 0
M 0
M 0
3D2D
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Compatibility Conditions
Relate the deformations of a structure so that its various parts fit together without any gaps or overlaps.
Ensure that the deformed shape of the structure is continuous and consistent with the support conditions.
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Constitutive relations
Describe the relationships between the stresses and strains of a structure in accordance with the stress-strain properties of the structural material
Provide the link between the equilibrium equations and compatibility conditions
Linear elastic constitutive relations will be used in this course
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Constitutive relations
Stress-strain curve for low-carbon steel. Hooke's law is only valid for the portion of the curve between the origin and the yield point.
1. Ultimate strength
2. Yield strength-corresponds to yield point.
3. Rupture
4. Strain hardening region
5. Necking region.
Example:
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Constitutive relations
and in tensor form,
or, equivalently,
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Linear vs. Nonlinear Analysis
In this course we focus on linear analysis:
Assumptions of linear analysis:
1- The structures are composed of linearly elastic materials
2- The deformations of the structures are so small
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Software
Software for the analysis of framed structures using matrix stiffness method is provided on the CD-ROM included with the textbook.
The software can be used to verify the correctness of Problems in Assignments and in-class activities and to visualize the deformed shapes of structures
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End of Lecture 1 Module 1