MATRIX ANALYSIS OF STRUCTURES (CE854)

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Matrix Analysis of Structures

Assignment No. 1 (Matrix Algebra)

Introduction:

Analysis of statically indeterminate (in some cases determinate) structures

generally requires the solution of linear simultaneous equations, the number of

which depends on the method of analysis. Some methods avoid simultaneous

equations by using iterative or successive correction techniques in order to

reduce the amount of computation, (e.g. moment distribution method) and are

suitable when the calculations are made by hand or by a hand-held or small

desk calculator.

For large and complicated structures hand computations is often

impracticable, and a digital computer has to be used. Its advent has shifted the

emphasis from easy problem solution to efficient problem formulation: using

matrices and matrix algebra, a large quantity of information can be organized

and manipulated in a compact form.

In this course we will study the basic computer methods but not the details

of programming.

It is emphasized that the hand methods of solution must not be neglected.

They are of value not only when a computer is not available but also for

preliminary calculations and for checking of computer results.

Course Content:

Course Concept

Capability to Understand the Behavior of Framed Structures

Understanding of Fundamental Concepts in Matrix (Computer) Analysis of

Structures

Analysis of Three Dimensional Structures

Ability to Analyse Complex Framed Structures Using a Computer

MATRIX ANALYSIS OF STRUCTURES (CE854)

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Review of Fundamentals in Structural Analysis

Structure, Types of Framed Structures

Structural Modeling or Structural Idealization

Supports and Reactions

Loads and Load Paths

Sign Convention of Forces & Displacements and Direction Cosines

Equilibrium of a Body

Statically Determinate Structures

Example Problems

Structure, Types of Framed Structures

“Any body that retains its physical form can be designated as a structure.”

Types of Framed Structures

o Continuous Beam

o Plane Truss

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o Plane Frame

o Horizontal Grid Subjected to Vertical Loads

o Space Truss

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o Space Frame

Examples of 3D Structures

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Structural Modeling or Structural Idealization

“Prismatic members are modeled as wire frames with lines passing through

the centerline of members.”

“Some shell and plate structures can be modeled as frames.”

o Culverts

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o Retaining Walls

o Building Structures etc.

Supports and Reactions

Plane Frames

o Hinge Support

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o Roller Support

o Fixed Support

o Encastre Roller

o Spring Supports

Space Frames

o Hinge or Roller Support

x

y

z

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o Fixed Support

o Spring Support

Loads and Load Paths

Loads

o Point loads

o Distributed Loads

o Couples or Moments

q/unit length

x

y

z

x

y

z

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o Temperature Gradients

o Construction Deficiencies and Miss-fits

o Shrinkage and Creep

o Support Settlement etc.

Δ T

MATRIX ANALYSIS OF STRUCTURES (CE854)

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Load Paths

Sign Convention for Forces & Displacements

Reference Axis

o Plane Structures

o Space Structures

Force/Displacement Direction in Plane Frames

x

y

z

2D Space

x

y

z

x

y

z

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Force/Displacement Direction in Space Frames

Equilibrium of a Body

“A body is said to be in equilibrium if the resultant of acting forces and

moments is zero in all directions.”

Equations of Equilibrium

o ∑ Fx = 0

o ∑ Fy = 0

o ∑ Fz = 0

o ∑ Mx = 0

o ∑ My = 0

o ∑ Mz = 0

x

y

z

3D Space

O

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Statically Determinate Structures

“A structure is said to be statically determinate if the forces can be found

from the equations of equilibrium alone”

Examples

o Simply Supported Beam

o Simply Supported Plane Frame etc.

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Problem No. L1-1

Find member forces and reaction components of the following truss.

3b

5b

5b

P P

P/5

A B

D C

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Problem No. L1-2

Find member forces and reaction components of the following truss.

3c

2c

5b

Top View Elevation

2b 2b

A

A B

B C C D D

2P

3P

5P 2P

2b 2b

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Problem No. L1-3

Find member forces and reaction components of the following beam.

0.4L

D B

0.6L 0.2L

C

A

qL 0.5qL

q/unit length

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Problem No. L1-4

Find member forces and reaction components of the following frame.

L L

L

P

P

A

B C

D

90o

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Problem No. L1-5

Find member forces and reaction components of the following frame.

L

B

0.15L

C A

q per unit length of horizontal projection

D

1

3 1

3