Intro to Matrix Analysis

8/20/2019 Intro to Matrix Analysis

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Engineering Systems

Lumped Parameter

(Discrete)Continuous

• A finite number ofstate variablesdescribe solution

• Algebraic Equations

• Differential EquationsGovern Response


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Lumped Parameter

Displacements of Joints fully describe sDisplacements of Joints fully describe s


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Matrix Structural Analysis - Objectives

Use

Equations of Equilibrium

Constitutive Equations

Compatibility Conditions

Basic Equations

Form

[A]{x}={b}

Solve for Unno!n "ispla#ements$For#es

{x}= [A]%&{b}


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Terminology

Element:Discrete StructuralMemberodes:Characteristicpoints that deneelement!"O"#":

All possibledirections ofdisplacements @ anode


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Assumptions

• 'inear Strain%"ispla#ement (elations)ip

• Small "eformations

• Equilibrium *ertains to Undeformed Confi+uration


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T$e Sti%%ness Met$od

&onsider a simple spring structural member

Undeformed Configuration

Deformed Configuration


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!erivation o% Sti%%ness Matrix 'sing (asicE)uations

1

2

*1

*2


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2

=

1+

For each case write basic equations


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2=0 1

P11 P21

X

Equilibrium

21112111 0 P P P P −=⇒=+Constitutive

111 δ k P =

P11

121 δ k P −=111 δ k P =


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2

1=0

P12P22

Equilibrium

12222212 0 P P P P −=⇒=+Constitutive

222 δ k P =

212 δ k P −=222 δ k P =


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Combined Action

P1P2

2112111 δ δ k k P P P −=+=

2122212 δ δ k k P P P +−=+=


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In atri! Form

2112111 δ δ k k P P P −=+=

2122212 δ δ k k P P P +−=+=

−

−

=

2

1

2

1

δ

δ

k k

k k

P

P


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&onsider * Springs

2 elements 3 nodes 3 dof

1

Fix Fix

2

Fix Fix

3

Fix Fix

+ * ,

+ *


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*-Springs

=

−

+−

−

3

2

1

3

2

1

22

2

2

1

1

11

0

0

P

P P

k k

k k

k k

k k

δ

δ δ

&ompare to +-Spring

=

−

−2

1

2

1

11

11

P

P

k k

k k

δ

δ


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Properties o% Sti%%ness Matrix

SM is Symmetric(etti-Max.ell La.

SM is Singularo (oundary &onditions Applied /et

Main !iagonal o% SM Positiveecessary %or Stability


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Apply (oundary &onditions

k ii k ij k ik k il k im uiu j

uk

ul

k ji k jj k jk k jl k jm

k ki k kj k kk k kl k km

k li k lj k lk k ll k lm

k li k lj k lk k ll k lm um

!i! j

!k

!l

!m

" ##

" #s

" s# " ss

u#

P#

us Ps

" ff uf " " fsusPf

" sf uf " " ssusPs " sf uf " " ssusPs

uf " ff #Pf " " fsus$$1


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Trans%ormations

P

k 2k 1

δ1

δ2

δ3

δ2

u1

u2

u3

u%

u3

u%

u&

u'

x

(Global C$

xLocal C$

b&ective' ransform $tate ariables from LC$ to GC$


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Trans%ormations

Consider

P*"

P*+

P*+ , P*+cosφ + P*"sinφ

P*" , -P*+sinφ + P*"cosφ

P*+

P*",

cosφ sinφ

-sinφ cosφ

P*+

P*"

P* , P*

x

(Global C$

φ

P*+

P.+

.

P."

*


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Trans%ormations

/n General/n General

P*"

P*+ P* , P*

x

(Global C$

φ

P*+

P.+

.

P."

*

P. , P.

u. , u.

u* , u*

$imilarl" for u$imilarl " for u

P* , P*-*

or

P. , P.-*

or


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Trans%ormationsElement stiffness equations in Local C$

0 ,* -*

-* *

δ*δ.

P*

P.

E+pand to 1 Local dof

0

* # -* ## # # #-* # * ## # # #

u*+u*"

u.+u."

,

P*+P*"

P.+P."

P*+

P.+P."

P*"

φ

.

*

P*

P.


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SM in 0lobal &oordinate System

/ntroduce t2e transformed variables3

4 u , PRR-*

4 ' Element $% in global C$

56 5#6

5#6 565R6,7ot2 R and Depend on Particular Element


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Trans%ormations

!or e+ample for an a+ial element 8it2 0,AE9L

AE9L

l. lm - l. - lm

$"mm:

m. - lm - m.

l. lmm.

4 ,

l,cosφ m,sinφ


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1n Summary

• !erivation o% element SM 2 (asicE)uations

• Structural SM by Superposition

• Application o% (oundary &onditions -

Elimination

• Solution o% Sti%%ness E)uations 2

Partitioning

• Local 3 0lobal &S

• Trans%ormation

Intro to Matrix Analysis

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