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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